Chapter 1.2, Problem 65E

(a)

To determine

To simplify: the given expression ${\left(8a^6{b}^{\frac{3}{2}}\right)}^{\frac{2}{3}}$ .

Answer to Problem 65E

  ${\left(8a^6{b}^{\frac{3}{2}}\right)}^{\frac{2}{3}}=4a^{4}b$

Explanation of Solution

Given information:

Given expression

  ${\left(8a^6{b}^{\frac{3}{2}}\right)}^{\frac{2}{3}}$

Calculation:

Consider the given expression.

  ${\left(8a^6{b}^{\frac{3}{2}}\right)}^{\frac{2}{3}}$

Now, use the law of exponents, which states

  ${\left(a^m\right)}^n=a^{mn}$ and ${\left(ab\right)}^n=a^n{b}^n$

And combine like terms

  $\begin{array}{c}{\left(8a^6{b}^{\frac{3}{2}}\right)}^{\frac{2}{3}}=8^{\frac{2}{3}}\cdot {\left(a^6\right)}^{\frac{2}{3}}\cdot {\left(b^{\frac{3}{2}}\right)}^{\frac{2}{3}}\\ ={\left(2^3\right)}^{\frac{2}{3}}\cdot {\left(a^6\right)}^{\frac{2}{3}}\cdot {\left(b^{\frac{3}{2}}\right)}^{\frac{2}{3}}\\ ={\left(2\right)}^{3\times \left(\frac{2}{3}\right)}\cdot {\left(a\right)}^{6\times \frac{2}{3}}\cdot {\left(b\right)}^{\frac{3}{2}\times }{}^{\frac{2}{3}}\\ =2^2\cdot a^4\cdot b\\ =4a^{4}b\end{array}$

Hence,

  ${\left(8a^6{b}^{\frac{3}{2}}\right)}^{\frac{2}{3}}=4a^{4}b$

(b)

To determine

To simplify: the given expression ${\left(4a^6{b}^8\right)}^{\frac{3}{2}}$ .

Answer to Problem 65E

  ${\left(4a^6{b}^8\right)}^{\frac{3}{2}}=8a^9{b}^{12}$

Explanation of Solution

Given information:

Given expression

  ${\left(4a^6{b}^8\right)}^{\frac{3}{2}}$

Calculation:

Consider the given expression.

  ${\left(4a^6{b}^8\right)}^{\frac{3}{2}}$

Now, use the law of exponents, which states

  ${\left(a^m\right)}^n=a^{mn}$ and ${\left(ab\right)}^n=a^n{b}^n$

And combine like terms

  $\begin{array}{c}{\left(4a^6{b}^8\right)}^{\frac{3}{2}}=4^{\frac{3}{2}}\cdot {\left(a^6\right)}^{\frac{3}{2}}\cdot {\left(b^8\right)}^{\frac{3}{2}}\\ ={\left(2^2\right)}^{\frac{3}{2}}\cdot {\left(a^6\right)}^{\frac{3}{2}}\cdot {\left(b^8\right)}^{\frac{3}{2}}\\ ={\left(2\right)}^{2\times \left(\frac{3}{2}\right)}\cdot {\left(a\right)}^{6\times \frac{3}{2}}\cdot {\left(b\right)}^{8\times }{}^{\frac{3}{2}}\\ =2^3\cdot a^9\cdot b^{12}\\ =8a^9{b}^{12}\end{array}$

Hence,

  ${\left(4a^6{b}^8\right)}^{\frac{3}{2}}=8a^9{b}^{12}$