Chapter 1.5, Problem 3E

(a)

To determine

To simplify: The expression $b^8{\left(2b\right)}^4$, use law of exponents.

Answer to Problem 3E

The expression $b^8{\left(2b\right)}^4=16b^{12}$.

Explanation of Solution

Law of exponents:

If x and y are positive numbers and m and n are real numbers, then

$\begin{array}{l}a.x^{m+n}=x^m{x}^n\\ b.x^{m-n}=\frac{x^m}{x^n}\\ c.{\left(x^m\right)}^n=x^{mn}\\ d.{\left(xy\right)}^m=x^m{y}^n\end{array}$

Calculation:

Use the law ${\left(xy\right)}^m=x^m{y}^n$, and rewrite the expression as $b^8{\left(2b\right)}^4=b^8{\left(2\right)}^4{\left(b\right)}^4$.

Use the law $x^{m+n}=x^m{x}^n$, and simplify as follows.

$\begin{array}{c}{b}^8{\left(2\right)}^4{\left(b\right)}^4=b^8{\left(b\right)}^4{\left(2\right)}^4\\ =b^{8+4}\left(16\right)\\ =16b^{12}\end{array}$

Thus, the given expression is simplified as $b^8{\left(2b\right)}^4=16b^{12}$.

(b)

To determine

To simplify: The expression $\frac{{\left(6y^3\right)}^4}{2y^5}$, use law of exponents.

Answer to Problem 3E

The expression $\frac{{\left(6y^3\right)}^4}{2y^5}=648y^7$.

Explanation of Solution

Law of exponents:

If x and y are positive numbers and m and n are real numbers, then

$\begin{array}{l}a.x^{m+n}=x^m{x}^n\\ b.x^{m-n}=\frac{x^m}{x^n}\\ c.{\left(x^m\right)}^n=x^{mn}\\ d.{\left(xy\right)}^m=x^m{y}^n\end{array}$

Calculation:

Use the law ${\left(xy\right)}^m=x^m{y}^n$, and rewrite the expression as $\frac{{\left(6y^3\right)}^4}{2y^5}=\frac{6^4{\left(y^3\right)}^4}{2y^5}$.

Use the law ${\left(x^m\right)}^n=x^{mn}$, and simplify such as, $\frac{6^4{\left(y^3\right)}^4}{2y^5}=\frac{1296y^{12}}{2y^5}$.

Then, use the law $x^{m-n}=\frac{x^m}{x^n}$, and simplify further as follows.

$\begin{array}{c}\frac{1296y^7}{2y^5}=648y^{12-5}\\ =648y^7\end{array}$

Thus, the given expression is simplified as $\frac{{\left(6y^3\right)}^4}{2y^5}=648y^7$.