5.2.1 Indices, PT3 Practice

5.2.1 Indices, PT3 Practice
Question 1:
(a) Simplify: a4 ÷ a7
(b)   Evaluate: ${\left({2}^{4}\right)}^{\frac{1}{2}}×{3}^{\frac{1}{2}}×{12}^{\frac{1}{2}}$

Solution:
(a) a4 ÷ a7 = a4-7 = a-3

(b)
$\begin{array}{l}{\left({2}^{4}\right)}^{\frac{1}{2}}×{3}^{\frac{1}{2}}×{12}^{\frac{1}{2}}={2}^{2}×{3}^{\frac{1}{2}}×{\left(4×3\right)}^{\frac{1}{2}}\\ \text{}={2}^{2}×{3}^{\frac{1}{2}}×{\left({2}^{2}×3\right)}^{\frac{1}{2}}\\ \text{}={2}^{2}×{3}^{\frac{1}{2}}×2×{3}^{\frac{1}{2}}\\ \text{}={2}^{3}×3\\ \text{}=24\end{array}$

Question 2:
(a) Simplify: p3 ÷ p-5
(b) Evaluate: ${10}^{\frac{1}{2}}×{5}^{-\frac{1}{2}}×{\left({2}^{\frac{1}{2}}\right)}^{5}$

Solution:
(a) p3 ÷ p-5 = p3-(-5) = p3+5 = p8

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

Question 3:
(a) Find the value of ${10}^{\frac{4}{3}}÷{10}^{\frac{1}{3}}.$
(b)   Simplify (xy3)5 × x4.

Solution:
(a)
$\begin{array}{l}{10}^{\frac{4}{3}}÷{10}^{\frac{1}{3}}\\ ={10}^{\frac{4}{3}-\frac{1}{3}}\\ ={10}^{\frac{3}{3}}\\ =10\end{array}$

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

Question 4:
(a) ${\left(81{a}^{8}\right)}^{-\frac{1}{4}}=$
(b)   Find the value of 23 × 22

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)
23 × 22 = 23+2 = 25 = 32

Question 5:
Find the value of the following.
(a) ${81}^{\frac{3}{4}}×{27}^{-1}$
(b) ${8}^{\frac{2}{3}}×{3}^{-2}$

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

(b)
$\begin{array}{l}{8}^{\frac{2}{3}}×{3}^{-2}={\left(\sqrt[3]{8}\right)}^{2}×\frac{1}{{3}^{2}}\\ \text{}={\left(2\right)}^{2}×\frac{1}{{3}^{2}}\\ \text{}=4×\frac{1}{9}\\ \text{}=\frac{4}{9}\end{array}$

Question 6:
Find the value of the following.
(a) ${16}^{\frac{4}{3}}×{\left({3}^{-2}\right)}^{3}×{9}^{\frac{3}{2}}$
(b) $\frac{{2}^{-2}×{3}^{2}}{{2}^{-3}×81}$

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

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

Question 7:
Find the value of the following.
(a)   (23)2 × 24 ÷ 25
(b) $\frac{{a}^{2}×{a}^{\frac{1}{2}}}{{\left({a}^{\frac{2}{3}}×{a}^{\frac{1}{3}}\right)}^{-2}}$

Solution:
(a)
(23)2 × 24 ÷ 25= 26 × 24 ÷ 25
= 26+4-5
= 25
= 32

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