Chapter 1.2, Problem 38E

(a)

To determine

To simplify: given expression $\frac{z^2{z}^4}{z^3{z}^{-1}}$ .

Answer to Problem 38E

  $\frac{z^2{z}^4}{z^3{z}^{-1}}=z^4$ .

Explanation of Solution

Given information:

Given expression

  $\frac{z^2{z}^4}{z^3{z}^{-1}}$

Calculation:

Consider the given expression.

  $\frac{z^2{z}^4}{z^3{z}^{-1}}$

To simplify this, use the product rule exponents, which states

  $\begin{array}{l}{a}^m{a}^n=a^{m+n}\\ \frac{z^2{z}^4}{z^3{z}^{-1}}=\frac{z^6}{z^2}\end{array}$

Now, use the second law of exponents, which states

  $\begin{array}{c}\frac{a^m}{a^n}=a^{m-n}\\ \frac{z^6}{z^2}=z^{6-2}\\ \frac{z^6}{z^2}=z^4\\ \frac{z^2{z}^4}{z^3{z}^{-1}}=z^4\end{array}$

(b)

To determine

To evaluate: the given expression ${\left(2y^2\right)}^3$ .

Answer to Problem 38E

  ${\left(2y^2\right)}^3=8y^6$

Explanation of Solution

Given information:

Given expression

  ${\left(2y^2\right)}^3$ .

Calculation:

Consider the given expression.

  ${\left(2y^2\right)}^3$

To simplify this, use the fourth law of exponents, which states

  $\begin{array}{l}{\left(ab\right)}^n=a^n{b}^n\\ {\left(2y^2\right)}^3={\left(2\right)}^3{\left(y^2\right)}^3\\ {\left(2y^2\right)}^3=8{\left(y^2\right)}^3\end{array}$

Now, use the third law of exponents, which states

  $\begin{array}{l}{\left(a^m\right)}^n=a^{mn}\\ 8{\left(y^2\right)}^3=8y^{2\cdot 3}\\ 8{\left(y^2\right)}^3=8y^6\\ {\left(2y^2\right)}^3=8y^6\end{array}$

(c)

To determine

To evaluate: given expression ${\left(8x\right)}^2$ .

Answer to Problem 38E

  ${\left(8x\right)}^2=64x^2$

Explanation of Solution

Given information:

Given expression

  ${\left(8x\right)}^2$

Calculation:

Consider the given expression.

  ${\left(8x\right)}^2$

To simplify this, use the fourth law of exponents, which states

  $\begin{array}{l}{\left(ab\right)}^n=a^n{b}^n\\ {\left(8x\right)}^2={\left(8\right)}^2{\left(x\right)}^2\\ {\left(8x\right)}^2=64x^2\end{array}$