Chapter 1.5, Problem 4E

Use the Law of Exponents to rewrite and simplify the expression.

4. (a) $\frac{x^{2n}\cdot x^{3n-1}}{x^{n+2}}$

(b) $\frac{\sqrt{a\sqrt{b}}}{\sqrt[3]{ab}}$

To determine

To simplify: The expression $\frac{x^{2n}\cdot x^{3n-1}}{x^{n+2}}$ using the law of exponents.

Answer to Problem 4E

The simplified form of the given expression is $\frac{x^{2n}\cdot x^{3n-1}}{x^{n+2}}=x^{4n-3}$.

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 given expression as,

$\begin{array}{c}\frac{x^{2n}\cdot x^{3n-1}}{x^{n+2}}=\frac{x^{2n+3n-1}}{x^{n+2}}\\ =\frac{x^{5n-1}}{x^{n+2}}\end{array}$

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

$\begin{array}{c}\frac{x^{5n-1}}{x^{n+2}}=x^{5n-1-\left(n+2\right)}\\ =x^{5n-1-n-2}\\ =x^{4n-3}\end{array}$

Thus, the given expression can be simplified as $\frac{x^{2n}\cdot x^{3n-1}}{x^{n+2}}=x^{4n-3}$.

To determine

To simplify: The expression $\frac{\sqrt{a\sqrt{b}}}{\sqrt[3]{ab}}$ using the law of exponents.

Answer to Problem 4E

The expression $\frac{\sqrt{a\sqrt{b}}}{\sqrt[3]{ab}}=\frac{a^{\frac{1}{6}}}{b^{\frac{1}{12}}}$.

Explanation of Solution

Use the rule $\sqrt[a]{x}=x^{\frac{1}{a}}$, and rewrite the expression as $\frac{{\left(a{b}^{\frac{1}{2}}\right)}^{\frac{1}{2}}}{{\left(ab\right)}^{\frac{1}{3}}}$.

Then, apply the law ${\left(xy\right)}^m=x^m{y}^n$, and rewrite the expression as follows.

$\begin{array}{c}\frac{{\left(a{b}^{\frac{1}{2}}\right)}^{\frac{1}{2}}}{{\left(ab\right)}^{\frac{1}{3}}}=\frac{{\left(a\right)}^{\frac{1}{2}}{\left(b^{\frac{1}{2}}\right)}^{\frac{1}{2}}}{{\left(a\right)}^{\frac{1}{3}}{\left(b\right)}^{\frac{1}{3}}}\\ =\frac{a^{\frac{1}{2}}{b}^{\frac{1}{4}}}{a^{\frac{1}{3}}{b}^{\frac{1}{3}}}\end{array}$

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

$\begin{array}{c}\frac{a^{\frac{1}{2}}{b}^{\frac{1}{4}}}{a^{\frac{1}{3}}{b}^{\frac{1}{3}}}=\frac{a^{\frac{1}{2}-\frac{1}{3}}}{b^{\frac{1}{3}-\frac{1}{4}}}\\ =\frac{a^{\frac{1}{6}}}{b^{\frac{1}{12}}}\end{array}$

Thus, the given expression can be simplified as $\frac{\sqrt{a\sqrt{b}}}{\sqrt[3]{ab}}=\frac{a^{\frac{1}{6}}}{b^{\frac{1}{12}}}$.