Chapter DT, Problem 1E

1. a. Evaluate the expression:

   (i) ${\left(\frac{16}{9}\right)}^{3/2}$

   (ii) $\sqrt[3]{\frac{27}{125}}$

2. b. Rewrite the expression using positive exponents only: ${(x^{-2}{y}^{-1})}^3$

To determine

To evaluate: The expression ${\left(\frac{16}{9}\right)}^{\frac{3}{2}}$.

Answer to Problem 1E

The value of the expression ${\left(\frac{16}{9}\right)}^{\frac{3}{2}}$ is $\underset{\_}{\frac{64}{27}}$.

Explanation of Solution

Result used:

Laws of Exponents:

For any real numbers a, b and n${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}\text{,}\left(b\ne 0\right)$.

Calculation:

The given expression is ${\left(\frac{16}{9}\right)}^{\frac{3}{2}}$.

Use Exponent law to evaluate the expression.

$\begin{array}{c}{\left(\frac{16}{9}\right)}^{\frac{3}{2}}=\frac{{16}^{\frac{3}{2}}}{9^{\frac{3}{2}}}\\ =\frac{16\cdot \sqrt{16}}{9\cdot \sqrt{9}}\\ =\frac{16\cdot 4}{9\cdot 3}\\ =\frac{64}{27}\end{array}$

Thus, the value of the expression ${\left(\frac{16}{9}\right)}^{\frac{3}{2}}$ is $\underset{\_}{\frac{64}{27}}$.

To determine

To evaluate: The expression $\sqrt[3]{\frac{27}{125}}$.

Answer to Problem 1E

The value of the expression $\sqrt[3]{\frac{27}{125}}$ is $\underset{\_}{\frac{3}{5}}$.

Explanation of Solution

The given expression is $\sqrt[3]{\frac{27}{125}}$.

Use Exponent law to evaluate the expression.

$\begin{array}{c}\sqrt[3]{\frac{27}{125}}={\left(\frac{27}{125}\right)}^{\frac{1}{3}}\\ =\frac{{27}^{\frac{1}{3}}}{{125}^{\frac{1}{3}}}\\ =\frac{3}{5}\end{array}$

Thus, the value of the expression $\sqrt[3]{\frac{27}{125}}$ is $\underset{\_}{\frac{3}{5}}$.

To determine

To rewrite: The expression ${\left(x^{-2}{y}^{-1}\right)}^3$ by using exponents.

Answer to Problem 1E

The expression ${\left(x^{-2}{y}^{-1}\right)}^3$ can be rewritten as $\underset{\_}{\frac{1}{x^6{y}^3}}$.

Explanation of Solution

Negative exponent:

If n is a positive integer, then $a^{-n}=\frac{1}{a^n},a\ne 0$.

Calculation:

The given expression is ${\left(x^{-2}{y}^{-1}\right)}^3$.

Use negative exponent law to evaluate the expression.

$\begin{array}{c}{\left(x^{-2}{y}^{-1}\right)}^3={\left(x^{-2}\right)}^3{\left(y^{-1}\right)}^3\\ =x^{\left(-2\right)\left(3\right)}{y}^{\left(-1\right)\left(3\right)}\\ =x^{-6}{y}^{-3}\\ =\frac{1}{x^6}\cdot \frac{1}{y^3}\left(\text{Use Negative exponent law}\right)\end{array}$

That is, ${\left(x^{-2}{y}^{-1}\right)}^3=\frac{1}{x^6{y}^3}$.

Thus, the value of the expression ${\left(x^{-2}{y}^{-1}\right)}^3$ is $\underset{\_}{\frac{1}{x^6{y}^3}}$.