How to Evaluate Integers Raised to Rational Exponents

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

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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)

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6. 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

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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.