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Know Your RootsHow can square roots and cube roots be used to solve expressions and equations?

Lesson 3 will have you explore square roots and cube roots as a means for solving simple equations involving variables with exponents. You will generate answers and classify your answers as either rational or irrational. Before you can use the square roots and cube roots, you need to understand the difference between the square root and squaring a number.18th Grade Math StandardM.8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational.

The 8th grade math standard addressed in this lesson include:

Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational.

2Rational Verses IrrationalRational Numbers

425100

Irrational Numbers

2 1.4142135 2.236067 3.141592

This lesson is designed to build upon your knowledge from Lesson 1 in Unit 1. In lesson 1, you were taught that all numbers that are not rational are called irrational. You were also introduced to find and use rational approximations of irrational numbers. Today, you will be introduced to one way irrational numbers are utilized to find solutions of square roots and cube roots when the solution is not a positive rational integer.

3Square Root Verses Squaring Root- something that is an origin or source4 - Read as the square root of 4

To find the 4 you need to ask What number times itself equals 4? The answer is 2 2 squared, 2 times 2, is equal to 4. Therefore the 4 = 2 Remember, 2 is a rational number.

People often confuse squaring a number and taking the square root of a number. The dictionary defines the word root as: something that is an origin or source. The square root or cube root of a number is the original source before it is squared or cubed.

For example, this expression ( 4) is read as the square root of 4.

What number times itself equals 4? The answer is 22 squared, which is 2 times 2, is equal to 4. Therefore the 4 = 2 because 2 is the root of the number 4, it is the original source multiplied by itself to equal 4.

If you understand the word root to be the origin, then the confusion of squaring a number verses taking the square root of a number will be an easy task!

4Perfect Squares = Easy Square RootsIt is sometimes easier to find the square root of a perfect square.

? X ? = 2For example;25 = 5 because 5 x 5 = 25What do you do if you need to find the 2? 2 1.414213

It is sometimes easier to find the square root of a perfect square.

However, you will encounter a problem when you cannot find a rational value for finding the square root of a number. You will need to estimate or use a calculator to find the square root of a number.

What do you do if you need to find the 2, what number times itself equals 2? Based on your mental math abilities, it is hard to calculate an exact answer because there is not a rational solution, meaning there is not an integer that you could multiply by itself to get 2 exactly. This is where lesson 1 from this unit has helped you understand how to get the answer. By understanding that all numbers have a decimal expansion, you know answers are not always in integer form. There is not an exact number times itself that equals 2, so you have to estimate or use a calculator to find the value. This causes us to use irrational numbers to find solutions to our problems. The 2 is approximated at 1.414213 because it is an irrational number, the decimal expansion does not repeat and it does not terminate.

5Cube Root Verses Cubing Root- something that is an origin or source8 - Read as the cube root of 8

To find the 8 you need to ask ??? equals 8? The answer is 2 2 times 2 times 2 equals 8. Therefore the 8 = 2 Remember, 2 is a rational number.333

Lets try this concept with cube roots, once again people often confuse cubing a number and taking the cube root.

Remember, the dictionary defines the word root as: something that is an origin or source. The square root or cube root of a number is the original source before it is squared or cubed.

This expression (38) is read as the cube root of 8. What number cubed, a number times itself times itself again, equals 8?

The answer is 22 times 2 times 2 equals 8. Therefore the 38 = 2 because 2 is the root of the number 8, it is the original source, when you have 2 cubed, you multiply 2 three times to equal 8.6Lesson SummaryUse your exponent knowledge to square and cube numbers.Use square roots and cube roots to find solutions to equations.Classify numbers as either rational or irrational.

Upon completion of this lesson, you will have:

Use your exponent knowledge from lesson 2 in this unit to square and cube numbers.

Used square root and cube root symbols to find solutions to equations.

Classify numbers as either rational or irrational practicing you knowledge from lesson 1 in this unit.

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