Rational Exponents

Rational Exponents

How to simplify the numbers with rational exponents: definition, examples, and their solutions.

Definition

The nth root means the exponent 1/n. So the nth root of a^m is a^(m/n).

The nth root means the exponent 1/n.

So nam = am/n.

nth Root

Example 1: Rational Exponent of 316

Write the given radical using a rational exponent. Cube root [16]

16 = 24

324 = 24/3

The radical is a cube root.
It's an odd root.

So you don't have to think about
the absolute value sign.

nth Root

So 24/3 is the answer.

Example 2: Radical Form of a7/9

Write the given expression in radical form. a^(7/9)

a7/9 = 9a7

Example 3: Simplify 45/2

Simplify the given expression. 4^(5/2)

4 = 22

So (given) = 22⋅(5/2).

Power of a power

2⋅(5/2) = 5

25 = 32

So (given) = 32.

Example 4: Simplify 8-7/3⋅85/3

Simplify the given expression. 8^(-7/3) * 8^(5/3)

8-7/3⋅85/3 = 8-7/3 + 5/3

Product of powers

-7/3 + 5/3 = -2/3

8 = 23

So (given) = 23⋅(-2/3).

3⋅(-2/3) = -2

2-2 = 1/22

Negative exponent

22 = 4

So (given) = 1/4.

Example 5: Simplify x-5/6

Simplify the given expression. x^(-5/6)

x-5/6 = 1/x5/6

Negative exponent

The denominator is x5/6,
which is an irrational number.
(x5/6 = 6x5)

x5/6x1/6 = x

So, to rationalize the denominator,
multiply x1/6/x1/6.

Rationalizing a denominator

x5/6x1/6 = x5/6 + 1/6
= x6/6

Product of powers

x6/6 = x1
= x

So (given) = x1/6/x.

The given expression has a rational exponent.
So the answer should be using a rational exponent.

This means
x1/6/x is the answer
but 6x/x cannot be the answer.

Example 6: Simplify (3x26x)/x

Simplify the given expression. (Cube root [x^2] * 6th root [x]) / (Square root [x])

To combine these radicals,
write the radicals using rational exponents.

3x2 = x2/3

6x = x1/6

x = x1/2

The powers have the same base: x.
So combine the exponents.

x1/6 is multiplied.
So write the exponent +1/6.

Product of powers

x1/2 is divided.
So write the exponent -1/2.
(The sign is (-).)

Quotient of powers

2/3 = 4/6
1/2 = 3/6

So (given) = x4/6 + 1/6 - 3/6.

4/6 + 1/6 - 3/6 = 2/6

So (given) = x2/6.

2/6 = 1/3

The given expression is in radical form.
So the answer should be in radical form.

So change x1/3 in radical form.

x1/3 = 3x

So 3x is the answer.

Example 7: Simplify (x1/2 - 3)/(x1/2 + 1)

Simplify the given expression. [x^(1/2) - 3]/[x^(1/2) + 1]

Given: (x1/2 - 3)/[x1/2 + 1]

[x1/2 + 1] means [√x + 1].

So the denominator [x1/2 + 1] is an irrational binomial.

The conjugate of [x1/2 + 1] is [x1/2 - 1].

So, to rationalize the denominator,
multiply (x1/2 - 1)/(x1/2 - 1).

Rationalizing a denominator

(x1/2 + 1)(x1/2 - 1) = (x1/2)2 - 12

Product of a sum and a difference (a + b)(a - b)

Think x1/2 as a variable
and expand the numerator.

(x1/2 - 3)(x1/2 - 1)
= x - x1/2 - 3x1/2 + 3

FOIL method

(x1/2)2 = x

-x1/2 - 3x1/2 = -4x1/2

So (given) = (x - 4x1/2 + 3)/(x - 1).

The given expression has a rational exponent.
So the answer should be using a rational exponent.
So this is the answer.