6.2 Apply Properties of Rational Exponents

VOCABULARY

Simplest form of a radical

A radical with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized.

Like radicals

Two radical expressions with the same index and radicand.

PROPERTIES OF RATIONAL EXPONENTS

Let a and b be real numbers and let m and n be rational numbers. The following properties have the same names as those in Lesson 5.1, but now apply to rational exponents.

Property

1. a^m · a^n = a^m ^+ n	4^1/2 · 4^3/2 = 4^(1/2 + 3/2) _= 4^2 = 16_
2. (a^m)^n = a^m n	(2^5/2)^2 = 2^(5/2·^2) _= 2^5 = 32_
3. (ab)^m = a^mb^m	(16 · 4)^1/2 = 16^1/2 · 4^1/2 _= 4 · 2 = 8_
4. a^-m = , a ¹ 0	25^-1/2 = =
5. = a^m^- n a ¹ 0	= 3^(5/2^- 1/2) =_3^2 = 9_
6. = , b ¹ 0	= =

Example 1

Use properties of exponents

Use the properties of rational exponents to simplify the expression.

a. 9^1/2 · 9^3/4 = _9^{(1/2 + 3/4)} = 9^5/4_

b. (7^2/3 · 5^1/6)³ = _(7^2/3)³ · (5^1/6)³_

= _7^(2/3 · ³⁾ · 5^(1/6 · ³⁾_

= _7² · 5^1/2 = 49 · 5^1/2_

c. = _3^{(5/6 - 1/3)} = 3^3/6 = 3^1/2_

d. = = (4^2/3)4 = 4^{(2/3 · 4)} = 4^8/3

PROPERTIES OF RADICALS

Product Property of Radicals	Quotient Property of Radicals
	, b ¹ 0

Example 2

Use properties of radicals

Use the properties of radicals to simplify the expression.

a. Product property

b. Quotient property

Simplify the expression.

1. (6⁶ · 5⁶)^-1/6

2.

7

Example 3

Write the expression in simplest form.

Factor out perfect fifth power.

Product property

Simplify

Example 4

Add and subtract like radicals and roots

Simplify the expression.

a.

b.

Write the expression in simplest form.

3.

4.

Example 5

Simplify expressions involving variables

Simplify the expression. Assume all variables are positive.

a.

b. (36m⁴n¹⁰)^1/2 = _36^1/2(m⁴)^1/2(n¹⁰)^1/2_

= _6m^{(4· 1/2)}n^{(10 · 1/2)} = 6m²n⁵_

_{c. ____________}

d. =_7x^{(4 - 3/2)}y ^-(-3)z^{(7 - 5)} = 7x^5/2 y³z²_

Example 6

Write variable expressions in simplest form

Write the expression in simplest form. Assume all variables are positive.

Make denominator a perfect fourth power.

Simplify.

Quotient property.

Simplify.

Example 7

Add and subtract expressions involving variables

Perform the indicated operation. Assume all variables are positive.

a.

b. 3a²b^1/4 + 4a²b^1/4 = _(3 + 4)a² b^1/4 = 7a²b^1/4_

Simplify the expression. Assume all variables are positive.

5.

6.