Chapter 7.4, Problem 3P

a.

To determine

To write: the simplified form of expression ${\left(\frac{4}{x^3}\right)}^2$

Answer to Problem 3P

The required answer is: $\frac{16}{x^6}$

Explanation of Solution

Given information:

The given expression: ${\left(\frac{4}{x^3}\right)}^2$

Calculation:

  ${\left(\frac{4}{x^3}\right)}^2=\frac{4^2}{{(x^3)}^2}$

(Raise the numerator and denominator to second power)

  $=\frac{4^2}{x^{3\cdot 2}}$

(Multiply the exponents)

  $\begin{array}{l}=\frac{4^2}{x^6}\\ =\frac{16}{x^6}\end{array}$

(Simplify)

b.

To determine

To simplify: the expression ${\left(\frac{a^{\frac{3}{4}}}{a^5}\right)}^4$ in two different ways

Answer to Problem 3P

The required answer is: $\frac{1}{a^{17}}$

Explanation of Solution

Given information:

The given expression: ${\left(\frac{a^{\frac{3}{4}}}{a^5}\right)}^4$

Calculation:

First method:

  ${\left(\frac{a^{\frac{3}{4}}}{a^5}\right)}^4=\frac{\left(a^{\frac{3}{4}}\right)}{{(a^5)}^4}$

(Raise the numerator and denominator to fourth power)

  $=\frac{a^{\frac{3}{4}\cdot 4}}{a^{5\cdot 4}}$

(Multiply the exponents)

  $\begin{array}{l}=\frac{a^3}{a^{20}}\\ =a^{3-20}\end{array}$

(Subtract exponents when dividing powers with the same base)

  $\begin{array}{l}=a^{-17}\\ =\frac{1}{a^{17}}\end{array}$

(Rewrite using positive exponents)

Second method:

  ${\left(\frac{a^{\frac{3}{4}}}{a^5}\right)}^4={\left(a^{\frac{3}{4}-5}\right)}^4$

(Subtract exponents when dividing powers with the same base)

  $\begin{array}{l}={\left(a^{\frac{3-20}{4}}\right)}^4\\ =a^{\frac{-17}{4}\cdot 4}\end{array}$

(Multiply the exponents)

  $\begin{array}{l}=a^{-17}\\ =\frac{1}{a^{17}}\end{array}$

(Rewrite using positive exponents)

Second method is more preferable since there are fewer steps as compared to the first method.