# How to Evaluate Integers Raised to Rational Exponents

Evaluating integers raised to rational exponents involves using the properties of exponents and simplifying the expression.

## A step-by-step guide to evaluating integers raised to rational exponents

Here is a step-by-step guide to evaluating integers raised to rational exponents:

- Understand the problem: Read the expression carefully and identify the integer and the rational exponent.
- Write the expression in fractional form: Write the rational exponent as a fraction. For example, if the exponent is 2/3, write it as (2/3).
- Use the property of exponents: Apply the property that states that a^(m/n) = (a^m)^(1/n)
- Simplify the expression: Use the property that states that (a^m)^(1/n) = ∛a^m to simplify the expression.
- Evaluate the exponent: Evaluate the fractional exponent by finding the nth root of the integer.
- Write the final answer in a complete sentence.

### Evaluating integers raised to rational exponents**– Example** 1

Evaluate (-2)^(4/3)

**Solution:**

- We are given the expression (-2)^(4/3)
- We write the exponent in the fractional form: (-2)^(4/3)
- Apply the property a^(m/n) = (a^m)^(1/n) => (-2)^4 * (-2)^(1/3)
- Apply the property (a^m)^(1/n) =∛a^m => ∛(-2)^4 * ∛(-2)^(1/3) 5.
- Evaluate the exponent: ∛(-2)^4 = ∛(-16) = -2, since ∛(-16) = -4 and ∛(-2) = -2^(1/3)

- The final answer is (-2)^(4/3) = -2 * ∛(-2) = -2 * -2 = 4

So, (-2)^(4/3) = 4

It’s worth noting that when an integer is raised to a fractional exponent it becomes a complex number, and this method is for evaluating real numbers. Also, when an integer is raised to a fractional exponent, it is important to be aware of the properties of the nth roots and the properties of the base number, as some base numbers have multiple solutions.

### Evaluating integers raised to rational exponents**– Example** 2

Evaluate: (4^(2/3))^3

**Solution: **This question is asking you to simplify the expression by using the properties of exponents and simplify the fractional exponent.

To solve this question, we can use the following steps:

- Understand the problem: The expression is (4^(2/3))^3, which is 4 raised to the 2/3 power raised to the 3rd power.
- Use the property of exponents: Apply the property that states that (a^b)^c = a^(bc) to simplify the expression. This means that (4^(2/3))^3 = 4^(2/3 * 3)
- Simplify the exponent: 2/3 * 3 = 2
- Evaluate the exponent: 4^2 = 16
- Write the final answer in a complete sentence: The final answer is (4^(2/3))^3 = 4^(2) = 16

So, (4^(2/3))^3 = 16

It’s worth noting that this is just one way to evaluate integers raised to rational exponents, and you can use different methods depending on the information provided in the problem.

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