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Radicals and Pythagorean Theorem
Radicals and Pythagorean TheoremFrom here on we need a scientific calculator. You may use the digital calculators on your personal devices.
Radicals and Pythagorean TheoremFrom here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
Radicals and Pythagorean Theorem
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Radicals and Pythagorean Theorem
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Example A.
a. Sqrt(16) =c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =d. 3 =
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =d. 3 =
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3d. 3 =
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3d. 3 =
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3d. 3 = 1.732.. (calculator)
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3d. 3 = 1.732.. (calculator)
Note that the square of both +3 and –3 is 9, but we designate sqrt(9) or 9 to be +3.
Square Root
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.
“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3d. 3 = 1.732.. (calculator)
Note that the square of both +3 and –3 is 9, but we designate sqrt(9) or 9 to be +3. We say “–3” is the “negative of the square root of 9”.
From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.Square Root
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize.
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt of other small numbers using this table.
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36,
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36, so we estimate that30 5.5.
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36, so we estimate that30 5.5.In fact 30 5.47722….
Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25 = ±5
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25 = ±5
b. x2 = –4
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25 = ±5
b. x2 = –4 Solution does not exist.
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25 = ±5
b. x2 = –4 Solution does not exist.
c. x2 = 8
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25 = ±5
b. x2 = –4 Solution does not exist.
c. x2 = 8 x = ±8
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25 = ±5
b. x2 = –4 Solution does not exist.
c. x2 = 8 x = ±8 ±2.8284.. by calculator
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25 = ±5
b. x2 = –4 Solution does not exist.
c. x2 = 8 x = ±8 ±2.8284.. by calculator
exact answer approximate answer
(If c<0, there is no real solution.)
Equations of the form x2 = c has two answers:x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.a. x2
= 25 x = ±25 = ±5
b. x2 = –4 Solution does not exist.
c. x2 = 8 x = ±8 ±2.8284.. by calculator
exact answer approximate answerSquare-roots numbers show up in geometry for measuring distances because of the Pythagorean Theorem.
(If c<0, there is no real solution.)
A right triangle is a triangle with a right angle as one of its angle.
Radicals and Pythagorean Theorem
A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse,
Radicals and Pythagorean Theorem
hypotenuseC
A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, the two sides A and B forming the right angle are called the legs.
Radicals and Pythagorean Theorem
hypotenuse
legsA
BC
A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, the two sides A and B forming the right angle are called the legs. Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2 as shown
Radicals and Pythagorean Theorem
hypotenuse
legsA
BC
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw.a. a = 5, b = 12, c = ?
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw.a. a = 5, b = 12, c = ?
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw.a. a = 5, b = 12, c = ?
Since it is a right triangle,122 + 52 = c2
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw.a. a = 5, b = 12, c = ?
Since it is a right triangle,122 + 52 = c2
144 + 25 = c2
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw.a. a = 5, b = 12, c = ?
Since it is a right triangle,122 + 52 = c2
144 + 25 = c2
169 = c2
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw.a. a = 5, b = 12, c = ?
Since it is a right triangle,122 + 52 = c2
144 + 25 = c2
169 = c2
So c = ±169 = ±13
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw.a. a = 5, b = 12, c = ?
Since it is a right triangle,122 + 52 = c2
144 + 25 = c2
169 = c2
So c = ±169 = ±13Since length can’t be negative, therefore c = 13.
Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
b. a = 5, c = 12, b = ?Radicals and Pythagorean Theorem
b. a = 5, c = 12, b = ?Radicals and Pythagorean Theorem
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
Radicals and Pythagorean Theorem
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144
Radicals and Pythagorean Theorem
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25
Radicals and Pythagorean Theorem
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9.
Radicals and Pythagorean Theorem
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
The Distance Formula
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
The Distance Formula
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two points, D = distance between them,
The Distance Formula
(x1, y1)
(x2, y2)
D
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2
The Distance Formula
(x1, y1)
(x2, y2)
Δy
Δx
D
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where
The Distance Formula
(x1, y1)
(x2, y2)
Δx = x2 – x1
Δy
Δx = x2 – x1
D
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where
The Distance Formula
(x1, y1)
(x2, y2)
Δx = x2 – x1 Δy = y2 – y1and
Δy = y2 – y1
Δx = x2 – x1
D
by the Pythagorean Theorem.
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where
The Distance Formula
(x1, y1)
(x2, y2)
Δx = x2 – x1 Δy = y2 – y1and
Δy = y2 – y1
Δx = x2 – x1by the Pythagorean Theorem.Hence we’ve the Distant Formula:D = √ Δx2 + Δy2
D
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where
The Distance Formula
(x1, y1)
(x2, y2)
Δx = x2 – x1 Δy = y2 – y1and
Δy = y2 – y1
Δx = x2 – x1by the Pythagorean Theorem.Hence we’ve the Distant Formula:D = √ Δx2 + Δy2 = √ (x2 – x1)2 + (y2 – y1)2
D
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher Root
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher RootIf r3 = x, then we say a is the cube root of x.
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. 3
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, 3
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x.
3
k
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
3
k
k
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 =3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 23
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 =3 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –13 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 =3 3 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3 3 3 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 =
3 3 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2
3 3 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 =
3 3 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4 4
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 = not real
3 3 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4 4
Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)
D = (–3)2 + 72 = 58 7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 = not real f. 10 ≈ 2.15..
3 3 3
Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4 4 3
Radicals and Pythagorean TheoremRational and Irrational Numbers
The number 2 is the length of the hypotenuse of the right triangle as shown.
Radicals and Pythagorean TheoremRational and Irrational Numbers
2 = 12 + 12
1
1
The number 2 is the length of the hypotenuse of the right triangle as shown.
Radicals and Pythagorean TheoremRational and Irrational Numbers
2 = 12 + 12
1
1
It can be shown that 2 is not a fraction like 3/5, ½ etc..
The number 2 is the length of the hypotenuse of the right triangle as shown.
Radicals and Pythagorean TheoremRational and Irrational Numbers
2 = 12 + 12
1
1
It can be shown that 2 is not a fraction like 3/5, ½ etc.. i.e. 2 ≠ P/Q of any two integers P and Q.
The number 2 is the length of the hypotenuse of the right triangle as shown.
Radicals and Pythagorean TheoremRational and Irrational Numbers
2 = 12 + 12
1
1
It can be shown that 2 is not a fraction like 3/5, ½ etc.. i.e. 2 ≠ P/Q of any two integers P and Q.Therefore we said that2 is an irrational (non–ratio) number.
The number 2 is the length of the hypotenuse of the right triangle as shown.
Radicals and Pythagorean TheoremRational and Irrational Numbers
2 = 12 + 12
1
1
It can be shown that 2 is not a fraction like 3/5, ½ etc.. i.e. 2 ≠ P/Q of any two integers P and Q.Therefore we said that2 is an irrational (non–ratio) number. Most real numbers are irrational.
The number 2 is the length of the hypotenuse of the right triangle as shown.
Radicals and Pythagorean TheoremRational and Irrational Numbers
2 = 12 + 12
1
1
It can be shown that 2 is not a fraction like 3/5, ½ etc.. i.e. 2 ≠ P/Q of any two integers P and Q.Therefore we said that2 is an irrational (non–ratio) number. Most real numbers are irrational.
The real line is populated sparsely by fractional numbers. The Pythagorean school of the ancient Greeks had believed that all the measurable quantities in the universe are fractional quantities. The “discovery” of these extra irrational numbers caused a profound intellectual crisis.
The number 2 is the length of the hypotenuse of the right triangle as shown.
Radicals and Pythagorean TheoremRational and Irrational Numbers
2 = 12 + 12
1
1
It can be shown that 2 is not a fraction like 3/5, ½ etc.. i.e. 2 ≠ P/Q of any two integers P and Q.Therefore we said that2 is an irrational (non–ratio) number. Most real numbers are irrational.
The real line is populated sparsely by fractional numbers. The Pythagorean school of the ancient Greeks had believed that all the measurable quantities in the universe are fractional quantities. The “discovery” of these extra irrational numbers caused a profound intellectual crisis. It wasn’t until the last two centuries that mathematicians clarified the strange questions “How many and what kind of numbers are there?”
Radicals and Pythagorean TheoremExercise A. Solve for x. Give both the exact and approximate answers. If the answer does not exist, state so. 1. x2 = 1 2. x2 – 5 = 4 3. x2 + 5 = 4 4. 2x2 = 31 5. 4x2 – 5 = 4 6. 5 = 3x2 + 1 7. 4x2 = 1 8. x2 – 32 = 42 9. x2 + 62 = 102
10. 2x2 + 7 = 11 11. 2x2 – 5 = 6 12. 4 = 3x2 + 5
x
3
4
Exercise B. Solve for x. Give both the exact and approximate answers. If the answer does not exist, state so.
13. 4
3
x14. x
12
515.
x
1
116. 2
1
x17. 3 2
3
x18.
Radicals and Pythagorean Theorem
x
4
19.
x
x20.
3 /3
21.43 5 2 6 /3
Exercise C. Given the following information, find the rise and run from A to B i.e. Δx and Δy. Find the distance from A to B.
A22.
B
A
23.
B
24. A = (2, –3) , B = (1, 5) 25. A = (1, 5) , B = (2, –3)26. A = (–2 , –5) , B = (3, –2) 27. A = (–4 , –1) , B = (2, –3)28. Why is the distance from A to B the same as from B to A?
Exercise D. Find the exact answer.Radicals and Pythagorean Theorem
3–129. 30. 13
–12534. 8 31. –13
83–2732.
33. –13
643
10035. 100036. 3
10,00037. 1,000,00038. 3
0.0139. 0.00140. 3
0.000141. 0.00000142. 3
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