Simplifying

When simplifying a radical expression, find the factors that are to the nth powers of the radicand and then use the Product Property of Radicals.

What is the Product Property of Radicals???

Product Property of Radicals

abn = an ⋅ bn

For any real numbers a and b, and any integer n, n>1,

1. If n is even, then When a and b are both nonnegative.

2. If n is odd, then abn = an ⋅ bn

Let’s do a few problems together.

1. 144x8y5

12x4y2 y

2) 7 64n23 ⋅4 8n3 =

28⋅4⋅2⋅n=224n

Now, you try these examples.

1) 40x3

2) 56a6b9c11

3) 54x3y53

4) 16x2y ⋅ 3xy3

5) 27x35 ⋅ 9x2y65

Here are the answers:

1) 2x 10x

2) 2a3b4c5 14bc

3) 3xy 2y23

4) 4xy2 3x

5) 3xy y5

Quotient Property of Radicals

≠0,For real numbers a and b, b

And any integer n, n>1,

ab

n =an

bn , if all roots are defined.

81256

=81256

=916

Ex:

In general, a radical expression is simplified when:

The radicand contains no fractions.

No radicals appear in the denominator.(Rationalization)

The radicand contains no factors that are nth powers of an integer or polynomial.

Simplify each expression.

1) x6

y3

Rationalize the denominator

x3 yy2

Answer

To simplify a radical by adding or subtracting you must have like terms.

Like terms are when the powers AND radicand are the same.

Ex: 53 and 653 , 2x 6z and 5x 6z

Here is an example that we will do together.

3 20+ 150−5 45

3 22 ⋅5+ 52 ⋅6−5 32 ⋅5Rewrite using factors

3⋅2 5 +5 6−5⋅3 5

6 5 +5 6−15 5

−9 5+5 6 Combine like terms

Try this one on your own.

4 3+5 12−7 27

Answer: −7 3

You can add or subtract radicals like monomials. You can also simplify radicals by using the FOIL method of multiplying binomials.

Ex: (3 6 −2 3)(4+ 3)

Let us try one.

(3 6−2 3)(4+ 3)

12 6+9 2 −8 3−6

Since there are no like terms, you can not combine.

Lets do another one.

(8−5 3)(8+5 3)

−11

When there is a binomial with a radical in the denominator of a fraction, you find the conjugate and multiply. This gives a rational denominator.Ex: 5 +6 ⇒ Conjugate: 5−6

3−2 2 ⇒ Conjugate: 3+2 2

What is conjugate of 2 7+3?

Answer: 2 7 −3

Simplify: 5+65−3

5+65−3

=5+65−3

⋅5+35+3

Multiply by the conjugate.

5+3 5+6 5+185−9

FOIL numerator and denominator.

Next

23+9 5−4

Combine like terms

Try this on your own:

63+ 2

Answer: 3 6−2 3

7

Here are a mixed set of problems to do.

1) 540

2) 63 (4 123 −5 93 )

3) 84

9x34

4) 120−4 30

5) ( 7−2 2)( 6 +2 2)

6) 3− 54+ 3

Answers to the mixed set of problems.

1) 6 15

2) 8 93 −15 23

3) 72a4

3a

4) −2 30

5) 42+2 14−4 3−8

6) 12−3 3−4 5+ 15

13