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Chapter 3: Problem 8

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ (-27)^{4 / 3} $$

Short Answer

Expert verified
The expression \((-27)^{4/3}\) can be rewritten as \(\sqrt[3]{(-27)^4}\). Evaluating the power, we find that \((-27)^4 = 531441\). Now, we need to find the cube root of this value, which is \(81\). Thus, the final answer is \((-27)^{4/3} = 81\).

Step by step solution

01

Rewrite the fractional exponent as a power and a root

We will rewrite the expression as a power and a root: \[ (-27)^{4/3} = \sqrt[3]{(-27)^4} \]
02

Calculate the power

Evaluate the power inside the cube root: \[ (-27)^4 = (-27) \times (-27) \times (-27) \times (-27) \] Now since a negative number multiplied by a negative number results in a positive number, we can simplify this as follows: \[ (-27)^4 = (27 \times 27) \times (27 \times 27) \] Now, find the products of the numbers to get the result of the power: \[ (-27)^4 = 729 \times 729 \] which results in: \[ (-27)^4 = 531441 \]
03

Calculate the cube root

Now that we have the result of the power, we want to find the cube root of that value: \[ \sqrt[3]{531441} \] To find the cube root, we need to look for a number that, when multiplied by itself three times, gives us the number inside the cube root. We know that \(3 \times 3 \times 3 = 27\), which means \(9 \times 9 \times 9 = 729\). So it's logical to try the number 81: \[ 81 \times 81 \times 81 = 531441 \] Thus, the cube root of \(531441\) is \(81\): \[ \sqrt[3]{531441} = 81 \]
04

Write the final answer

Now that we have calculated the cube root, we can write the final answer: \[ (-27)^{4/3} = 81 \] Therefore, the value of \((-27)^{4/3}\) is \(81\).

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