Chapter 7: Chapter 7

Problem 102

Show that (a) \(\quad(-8)^{2 / 3}=\left(-8^{1 / 3}\right)^{2}\) (b) \(\quad(1 / 64)^{4 / 3}=\left[(1 / 64)^{1 / 3}\right]^{4}\)

Short Answer

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In summary, we have shown that the expressions are equal: (a) \(\quad(-8)^{2/3}=\left(-8^{1/3}\right)^{2}\), where both expressions simplify to 4. (b) \(\quad(1/64)^{4/3}=\left[(1/64)^{1/3}\right]^{4}\), where both expressions simplify to \(1/256\).

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Part (a): Simplify \((-8)^{2/3}\)

The given expression is \((-8)^{2/3}\). The exponent \(2/3\) can be interpreted as taking the cube root first and then squaring the result. So, let's simplify the expression step by step: \( \quad(-8)^{2/3} =[\,(-8)^{1/3}\,]^2 \) Now, calculate the cube root of \(-8\): \( \quad(-8)^{1/3} =-2 \) Now, square the result: \( \quad(-2)^2 =4 \)

Step 2: Part (a): Simplify \(\left(-8^{1/3}\right)^{2}\)

The given expression is \(\left(-8^{1/3}\right)^{2}\). First, find the cube root of -8: \( \quad-8^{1/3} =-2 \) Now, square the result: \( \quad(-2)^2 =4 \) Both expressions yielded the same result, so we can conclude that: \(\quad(-8)^{2/3} = \left(-8^{1/3}\right)^2 \)

Step 3: Part (b): Simplify \((1/64)^{4/3}\)

The given expression is \((1/64)^{4/3}\). The exponent \(4/3\) can be interpreted as taking the cube root first and then raising the result to the power of 4. So, let's simplify the expression step by step: \( \quad(1/64)^{4/3} =[\,(1/64)^{1/3}\,]^4 \) Now, calculate the cube root of \(1/64\): \( \quad(1/64)^{1/3} =1/4 \) Now, raise the result to the power of 4: \( \quad(1/4)^4 =1/256 \)

Step 4: Part (b): Simplify \(\left[(1/64)^{1/3}\right]^4\)

The given expression is \(\left[(1/64)^{1/3}\right]^4\). First, find the cube root of \(1/64\): \( \quad(1/64)^{1/3} = 1/4\) Now, raise the result to the power of 4: \( \quad(1/4)^4 = 1/256 \) Both expressions yielded the same result, so we can conclude that: \(\quad (1/64)^{4/3} = \left[(1/64)^{1/3}\right]^4\)

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Chapter 7: Chapter 7

Show that (a) \(\quad(-8)^{2 / 3}=\left(-8^{1 / 3}\right)^{2}\) (b) \(\quad(1 / 64)^{4 / 3}=\left[(1 / 64)^{1 / 3}\right]^{4}\)

Short Answer

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Step 1: Part (a): Simplify \((-8)^{2/3}\)

Step 2: Part (a): Simplify \(\left(-8^{1/3}\right)^{2}\)

Step 3: Part (b): Simplify \((1/64)^{4/3}\)

Step 4: Part (b): Simplify \(\left[(1/64)^{1/3}\right]^4\)

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