# 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:

1. Understand the problem: Read the expression carefully and identify the integer and the rational exponent.
2. 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).
3. Use the property of exponents: Apply the property that states that a^(m/n) = (a^m)^(1/n)
4. Simplify the expression: Use the property that states that (a^m)^(1/n) = ∛a^m to simplify the expression.
5. Evaluate the exponent: Evaluate the fractional exponent by finding the nth root of the integer.
6. Write the final answer in a complete sentence.

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

Evaluate (-2)^(4/3)

Solution:

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

1. Understand the problem: The expression is (4^(2/3))^3, which is 4 raised to the 2/3 power raised to the 3rd power.
2. 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)
3. Simplify the exponent: 2/3 * 3 = 2
4. Evaluate the exponent: 4^2 = 16
5. 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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