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Problem 102

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

Expert verified

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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Chapter 7

Simplify (a) \(^{3} \sqrt{-512}\) (b) \(^{4} \sqrt{(81 / 16)}\) (c) \(\left({ }^{3} \sqrt{-16}\right) /\left({ }^{3} \sqrt{-2}\right)\).

Chapter 7

Express with rational denominator $4 /\left({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3}+1\right)$.

Chapter 7

When \(\mathrm{x}=(3+5 \sqrt{-1}) / 2\), find the value of $2 \mathrm{x}^{3}+2 \mathrm{x}^{2}-7 \mathrm{x}+72$ and show that it will be unaltered if \((3-5 \sqrt{-1}) / 2\) be substituted for \(\mathrm{x}\).

Chapter 7

Find the product by inspection: \(\sqrt{3}(\mathrm{x}-\sqrt{5})(\mathrm{x}+\sqrt{5})\).

Chapter 7

Reduce \((2+3 \sqrt{-1})^{2} /(2+\sqrt{-1})\) to the form $\mathrm{A}+\mathrm{B} \sqrt{-1}$.

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