**5.2.1 Indices, PT3 Practice**

**Question 1:**

*a*

^{4 }÷

*a*

^{7}(a) Simplify:

${\left({2}^{4}\right)}^{\frac{1}{2}}\times {3}^{\frac{1}{2}}\times {12}^{\frac{1}{2}}$
(b) Evaluate:

Solution:Solution:

*a*

^{4 }÷

*a*

^{7 }=

*a*

^{4-7 }=

*a*

^{-3}(a)

(b)

**Question 2:**

*p*

^{3 }÷

*p*

^{-5}(a) Simplify:

${10}^{\frac{1}{2}}\times {5}^{-\frac{1}{2}}\times {\left({2}^{\frac{1}{2}}\right)}^{5}$
(b) Evaluate:

Solution:Solution:

*p*

^{3 }÷

*p*

^{-5 }=

*p*

^{3-(-5) }=

*p*

^{3+5 }=

*p*

^{8}

**(a)**

(b)

$\begin{array}{l}{10}^{\frac{1}{2}}\times {5}^{-\frac{1}{2}}\times {\left({2}^{\frac{1}{2}}\right)}^{5}\\ ={\left(2\times 5\right)}^{\frac{1}{2}}\times {5}^{-\frac{1}{2}}\times {2}^{\frac{5}{2}}\\ ={2}^{\frac{1}{2}}\times {5}^{\frac{1}{2}}\times {5}^{-\frac{1}{2}}\times {2}^{\frac{5}{2}}\\ ={2}^{\frac{1}{2}+\frac{5}{2}}\times {5}^{\frac{1}{2}+\left(-\frac{1}{2}\right)}\\ ={2}^{3}+{5}^{\frac{1}{2}-\frac{1}{2}}\\ ={2}^{3}+{5}^{0}\\ =8+1\\ =9\end{array}$
**Question 3:**

${10}^{\frac{4}{3}}\xf7{10}^{\frac{1}{3}}.$
(a) Find the value of

*xy*

^{3})

^{5}×

*x*

^{4}. (b) Simplify (

Solution:Solution:

(a)

$\begin{array}{l}{10}^{\frac{4}{3}}\xf7{10}^{\frac{1}{3}}\\ ={10}^{\frac{4}{3}-\frac{1}{3}}\\ ={10}^{\frac{3}{3}}\\ =10\end{array}$

$\begin{array}{l}\text{(b)}{\left(x{y}^{3}\right)}^{5}\times {x}^{4}={x}^{5}{y}^{15}\times {x}^{4}\\ \text{}={x}^{5+4}{y}^{15}\\ \text{}={x}^{9}{y}^{15}\end{array}$

**Question 4:**

${\left(81{a}^{8}\right)}^{-\frac{1}{4}}=$
(a)

^{3 }× 2

^{2}(b) Find the value of 2

Solution:Solution:

(a)

${\left(81{a}^{8}\right)}^{-\frac{1}{4}}=\frac{1}{{\left(81{a}^{8}\right)}^{\frac{1}{4}}}=\frac{1}{{\left({3}^{4}\right)}^{\frac{1}{4}}{\left({a}^{8}\right)}^{\frac{1}{4}}}=\frac{1}{3{a}^{2}}$

(b)

^{3 }× 2

^{2 }= 2

^{3+2 }= 2

^{5 }= 32

**Question 5:**

Find the value of the following.

${81}^{\frac{3}{4}}\times {27}^{-1}$
(a)

${8}^{\frac{2}{3}}\times {3}^{-2}$
(b)

Solution:Solution:

(a)

$\begin{array}{l}{81}^{\frac{3}{4}}\times {27}^{-1}={\left(\sqrt[4]{81}\right)}^{3}\times {\left({3}^{3}\right)}^{-1}\\ \text{}={\left(3\right)}^{3}\times {3}^{-3}\\ \text{}={3}^{3+\left(-3\right)}\\ \text{}={3}^{0}=1\end{array}$

(b)

**Question 6:**

Find the value of the following.

${8}^{\frac{4}{3}}\times {\left({3}^{-2}\right)}^{3}\times {9}^{\frac{3}{2}}$
(a)

$\frac{{2}^{-2}\times {3}^{2}}{{2}^{-3}\times 81}$
(b)

Solution:Solution:

**(a)**

$\begin{array}{l}{8}^{\frac{4}{3}}\times {\left({3}^{-2}\right)}^{3}\times {9}^{\frac{3}{2}}\\ ={\left({2}^{3}\right)}^{\frac{4}{3}}\times {3}^{-6}\times {\left({3}^{2}\right)}^{\frac{3}{2}}\\ ={2}^{4}\times {3}^{-6}\times {3}^{3}\\ =16\times {3}^{-6+3}\\ =16\times {3}^{-3}\\ =16\times \frac{1}{{3}^{3}}\\ =\frac{16}{27}\end{array}$

**(b)**

$\begin{array}{l}\frac{{2}^{-2}\times {3}^{2}}{{2}^{-3}\times 81}=\frac{{2}^{-2}\times {3}^{2}}{{2}^{-3}\times {3}^{4}}\\ \text{}={2}^{-2-\left(-3\right)}\times {3}^{2-4}\\ \text{}=2\times {3}^{-2}\\ \text{}=\frac{2}{{3}^{2}}\\ \text{}=\frac{2}{9}\end{array}$