Chapter 3.2, Problem 82E

Solution

(a)

The simplified form of the expression $\sqrt[4]{32x^4{y}^5}$ .

  $2xy\sqrt[4]{2y}$

Given:

The expression $\sqrt[4]{32x^4{y}^5}$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to simplify the given expression $\sqrt[4]{32x^4{y}^5}$ , use the exponent properties and combine the like terms to simplify further as shown below,

  $\begin{array}{c}\sqrt[4]{32x^4{y}^5}=\sqrt[4]{32}\cdot \sqrt[4]{x^4}\cdot \sqrt[4]{y^5}\\ =\sqrt[4]{16\cdot 2}\cdot x\cdot \sqrt[4]{y^4\cdot y}\\ =\sqrt[4]{16}\cdot \sqrt[4]{2}\cdot x\cdot \sqrt[4]{y^4}\cdot \sqrt[4]{y}\\ =\sqrt[4]{2^4}\cdot \sqrt[4]{2}\cdot x\cdot y\cdot \sqrt[4]{y}\\ =2xy\cdot \sqrt[4]{2\cdot y}\\ =2xy\sqrt[4]{2y}\end{array}$

Therefore, the simplified expression is $2xy\sqrt[4]{2y}$ .

(b)

The simplified form of the expression $\sqrt[5]{x^{10}{y}^5}$ .

  $x^{2}y$

Given:

The expression $\sqrt[5]{x^{10}{y}^5}$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to simplify the given expression $\sqrt[5]{x^{10}{y}^5}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}\sqrt[5]{x^{10}{y}^5}=\sqrt[5]{x^{10}}\cdot \sqrt[5]{y^5}\\ =\sqrt[5]{{\left(x^2\right)}^5}\cdot y\\ =x^2\cdot y\\ =x^{2}y\end{array}$

Therefore, the simplified expression is $x^{2}y$ .

(c)

The simplified form of the expression ${\left(8x^3{y}^7\right)}^{\frac{2}{3}}$ .

  $4x^2{y}^{\frac{14}{3}}$

Given:

The expression ${\left(8x^3{y}^7\right)}^{\frac{2}{3}}$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to simplify the given expression ${\left(8x^3{y}^7\right)}^{\frac{2}{3}}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{\left(8x^3{y}^7\right)}^{\frac{2}{3}}={\left(8\right)}^{\frac{2}{3}}\cdot {\left(x^3\right)}^{\frac{2}{3}}\cdot {\left(y^7\right)}^{\frac{2}{3}}\\ ={\left(2^3\right)}^{\frac{2}{3}}\cdot x^{3\cdot \frac{2}{3}}\cdot {\left(y^7\right)}^{\frac{2}{3}}\\ =2^{3\cdot \frac{2}{3}}\cdot x^{3\cdot \frac{2}{3}}\cdot y^{7\cdot \frac{2}{3}}\\ =2^2\cdot x^2\cdot y^{\frac{14}{3}}\\ =4x^2{y}^{\frac{14}{3}}\end{array}$

Therefore, the simplified expression is $4x^2{y}^{\frac{14}{3}}$ .

(d)

The simplified form of the expression ${\left(16x^9{y}^2\right)}^{\frac{3}{4}}$ .

  $8x^{\frac{27}{4}}{y}^{\frac{3}{2}}$

Given:

The expression ${\left(16x^9{y}^2\right)}^{\frac{3}{4}}$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to simplify the given expression ${\left(16x^9{y}^2\right)}^{\frac{3}{4}}$ , use the exponent properties and combine the like terms to simplify further as shown below,

  $\begin{array}{c}{\left(16x^9{y}^2\right)}^{\frac{3}{4}}={\left(16\right)}^{\frac{3}{4}}\cdot {\left(x^9\right)}^{\frac{3}{4}}\cdot {\left(y^2\right)}^{\frac{3}{4}}\\ ={\left(2^4\right)}^{\frac{3}{4}}\cdot x^{9\cdot \frac{3}{4}}\cdot y^{2\cdot \frac{3}{4}}\\ =2^3\cdot x^{\frac{27}{4}}\cdot y^{\frac{3}{2}}\\ =8x^{\frac{27}{4}}{y}^{\frac{3}{2}}\end{array}$

Therefore, the simplified expression is $8x^{\frac{27}{4}}{y}^{\frac{3}{2}}$ .