**Question 13:**

$\text{Solve:}{3}^{2n+1}=\frac{{3}^{n}\times 9}{{\left({9}^{\frac{1}{2}}\right)}^{-3}}$

*Solution*:$\begin{array}{l}{3}^{2n+1}=\frac{{3}^{n}\times 9}{{\left({9}^{\frac{1}{2}}\right)}^{-3}}\\ {3}^{2n+1}=\frac{{3}^{n}\times {3}^{2}}{{3}^{-3}}\\ {3}^{2n+1}={3}^{n+2-\left(-3\right)}\\ 2n+1=n+5\\ \text{}n=4\end{array}$

**Question 14:**

$\text{Giventhat}{k}^{3}={9}^{-\frac{3}{2}}\times {64}^{\frac{1}{2}},\text{findthevalueof}k.$

*Solution*:$\begin{array}{l}{k}^{3}={9}^{-\frac{3}{2}}\times {64}^{\frac{1}{2}}\\ \text{}={\left({3}^{2}\right)}^{-\frac{3}{2}}\times {\left({2}^{6}\right)}^{\frac{1}{2}}\\ \text{}={3}^{-3}\times {2}^{3}\\ \text{}={\left(\frac{2}{3}\right)}^{3}\\ k=\frac{2}{3}\end{array}$

**Question 15:**

${\text{Given9}}^{x+2}\xf7{3}^{4}={3}^{x+1},\text{calculatethevalueof}x.$

*Solution*:$\begin{array}{l}{\text{9}}^{x+2}\xf7{3}^{4}={3}^{x+1}\\ {\left({3}^{2}\right)}^{x+2}\xf7{3}^{4}={3}^{x+1}\\ \text{}{3}^{2x+4}\xf7{3}^{4}={3}^{x+1}\\ \text{}2x+4-4=x+1\\ \text{}2x=x+1\\ \text{}x=1\end{array}$

**Question 16:**

$\text{Simplify:}{\left(\frac{-2{x}^{5}{y}^{-2}}{{z}^{\frac{1}{6}}}\right)}^{3}\xf7\frac{1}{\sqrt{{x}^{2}z}}$

*Solution*:$\begin{array}{l}{\left(\frac{-2{x}^{5}{y}^{-2}}{{z}^{\frac{1}{6}}}\right)}^{3}\xf7\frac{1}{\sqrt{{x}^{2}z}}\\ =\frac{-8{x}^{15}{y}^{-6}}{{z}^{\frac{1}{2}}}\times \sqrt{{x}^{2}z}\\ =\frac{-8{x}^{15}{y}^{-6}}{{z}^{\frac{1}{2}}}\times {\left({x}^{2}z\right)}^{\frac{1}{2}}\\ =\frac{-8{x}^{15}{y}^{-6}}{\overline{){z}^{\frac{1}{2}}}}\times x\overline{){z}^{\frac{1}{2}}}\\ =-8{x}^{15+1}{y}^{-6}\\ =\frac{-8{x}^{16}}{{y}^{6}}\end{array}$

**Question 17:**

Find the value of the following.

^{3})

^{2 }× 2

^{4 }÷ 2

^{5}(a) (2

$\frac{{a}^{2}\times {a}^{\frac{1}{2}}}{{\left({a}^{\frac{2}{3}}\times {a}^{\frac{1}{3}}\right)}^{-2}}$
(b)

Solution:Solution:

**(a)**

(2

^{3})^{2 }× 2^{4 }÷ 2^{5}= 2^{6 }× 2^{4 }÷ 2^{5} = 2

^{6+4-5} = 2

^{5} =

**32****(b)**

$\begin{array}{l}\frac{{a}^{2}\times {a}^{\frac{1}{2}}}{{\left({a}^{\frac{2}{3}}\times {a}^{\frac{1}{3}}\right)}^{-2}}=\frac{{a}^{2+\frac{1}{2}}}{{\left({a}^{\frac{2}{3}}\times {a}^{\frac{1}{3}}\right)}^{-2}}\\ \text{}=\frac{{a}^{2+\frac{1}{2}}}{{\left({a}^{\frac{2}{3}+\frac{1}{3}}\right)}^{-2}}\\ \text{}=\frac{{a}^{\frac{5}{2}}}{{a}^{-2}}\\ \text{}={a}^{\frac{5}{2}-\left(-2\right)}\\ \text{}={a}^{\frac{5}{2}+\frac{4}{2}}\\ \text{}={a}^{\frac{9}{2}}\end{array}$