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Chapter 3: Problem 7

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ (-8)^{7 / 3} $$

Short Answer

Expert verified
The expression \((-8)^{7/3}\) can be rewritten as \(((-8)^{1/3})^7\). The cube root of -8 is -2, so the expression simplifies to \((-2)^7\). Multiplying the -2's together, we find that \((-8)^{7/3} = 128\).

Step by step solution

01

Rewrite the expression as a radical

Rewrite \((-8)^{7/3}\) as a cube root and a power: \(((-8)^{1/3})^7\).
02

Find the cube root of -8

Recall that the cube root of a number x is the number y such that \(y^3 = x\). In this case, the number inside the cube root is -8, and since \((-2)^3 = -8\), the cube root of -8 is -2. So, \((-8)^{1/3} = -2\).
03

Raise the result to the 7th power

We now have to compute \((-2)^7\). Using the exponent rules, we know that \((-2)^7 = -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2\).
04

Multiply the -2's together

Multiply the -2's together: \((-2)^7 = 128\). So, \((-8)^{7/3} = 128\).

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