Selina solutions concise maths class 9 icse Solutions for Chapter 7 chapter 7 indices exponents

Evaluate:

\begin{aligned} &\text { (i) }\\ &3^{3} \times(243)^{-\frac{2}{3}} \times 9^{\frac{1}{3}} \end{aligned}

\begin{aligned} &\text { (ii) }\\ &5^{-4} \times(125)^{5} \div(25)^{-\frac{1}{2}} \end{aligned}

\begin{aligned} &\text { (iii) }\\ &\left(\frac{27}{125}\right)^{\frac{2}{3}} \times\left(\frac{9}{25}\right)^{\frac{3}{2}} \end{aligned}

\begin{aligned} &\text { (iv) }\\ &7^{n} \times(25)^{-\frac{3}{2}}-5^{-3} \end{aligned}

\begin{aligned} &(\mathrm{v})\\ &\left(\frac{16}{81}\right)^{-\frac{3}{4}} \times\left(\frac{49}{9}\right)^{\frac{3}{2}}+\left(\frac{343}{216}\right)^{\frac{2}{3}} \end{aligned}

Simplify:

\begin{aligned} &\text { (i) }\\ &\left(8 x^{3} \div 125 y^{3}\right)^{\frac{2}{3}} \end{aligned}\begin{aligned} &\text { (ii) }\\ &(a+b)^{-1} \cdot\left(a^{-1}+b^{-1}\right) \end{aligned}\begin{aligned} &\text { (iii) }\\ &\frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^{n}-5^{n} \times 2^{2}} \end{aligned}\begin{aligned} &\text { (iv) }\\ &\left(3 x^{2}\right)^{-3} \times\left(x^{9}\right)^{\frac{2}{3}} \end{aligned}

Evaluate:

\begin{aligned} &(\mathrm{i})\\ &\sqrt{\frac{1}{4}}+(0.01)^{-\frac{1}{2}}-(27)^{\frac{2}{3}} \end{aligned} \begin{aligned} &\text { (ii) }\\ &\left(\frac{27}{8}\right)^{\frac{2}{3}}-\left(\frac{1}{4}\right)^{-2}+5^{0} \end{aligned}

Simplify each of the following and express with positive index:

\begin{aligned} &\text { (i) }\\ &\left(\frac{3^{-4}}{2^{-8}}\right)^{\frac{1}{4}} \end{aligned}

\begin{aligned} &\text { (ii) }\\ &\left(\frac{27^{-3}}{9^{-3}}\right)^{\frac{1}{5}} \end{aligned}

\begin{aligned} &\text { (iii) }\\ &(32)^{-\frac{2}{5}} \div(125)^{-\frac{2}{3}} \end{aligned}

\begin{aligned} &(\text { iv })\\ &\left[1-\left\{1-(1-n)^{-1}\right\}^{-1}\right]^{-1} \end{aligned}

\text { If } 2160=2^{a} \cdot 3^{b} \cdot 5^{c}, \text { find } a, b \text { and } c \text { . Hence calculate the value of } 3^{a} \times 2^{-b} \times 5^{-c}

\text { If } 1960=2^{a} \cdot 3^{b} \cdot 5^{c}, \text { find } a, b \text { and } c \text { . Calculate the value of } 2^{-a} \times 7^{b} \times 5^{-c}

Simplify:

(i) \frac{8^{3 a} \times 2^{5} \times 2^{2 a}}{4 \times 2^{11 a} \times 2^{-2 a}}

\begin{aligned} &\text { (ii) }\\ &\frac{3 \times 27^{n+1}+9 \times 3^{3 n-1}}{8 \times 3^{3 n}-5 \times 27^{n}} \end{aligned}

Show that:

\left(\frac{a^{m}}{a^{-n}}\right)^{m-n} \times\left(\frac{a^{n}}{a^{-k}}\right)^{n-1} \times\left(\frac{8^{e}}{a^{-m}}\right)^{f-m}

\text { If } a=x^{m+n} \cdot y^{\prime} ; b=x^{n+1} \cdot y^{m} \text { and } c=x^{1+m} \cdot y^{n}, \text { prove that: } a^{m-n} \cdot b^{n-1} \cdot c^{1-m}=1

Simplify:

\begin{aligned} &\text { (i) }\\ &\left(\frac{x^{a}}{x^{b}}\right)^{a^{2}+a b+b^{2}} \times\left(\frac{x^{b}}{x^{c}}\right)^{b^{2}+b o+c^{2}} \times\left(\frac{x^{c}}{x^{a}}\right)^{c^{2}+c a+z^{2}} \end{aligned}

\begin{aligned} &(\mathrm{ii})\\ &\left(\frac{x^{a}}{x^{-6}}\right)^{a^{2}-a b+b^{2}} \times\left(\frac{x^{b}}{x^{-c}}\right)^{b^{2}-b c+c^{2}} \times\left(\frac{x^{c}}{x^{-a}}\right)^{c^{2}-c a+z^{2}} \end{aligned}

Solve for x:

\text { (i) } 2^{2 x+1}=8\begin{aligned} &\text { (ii) }\\ &2^{5 x-1}=4 \times 2^{3 x+1} \end{aligned}\begin{aligned} &\text { (iii) }\\ &3^{4 x+1}=27^{(x+1)} \end{aligned}\begin{aligned} &(\text { iv })\\ &49^{x+4}=7^{2}(343)^{(x+1)} \end{aligned}

Find x, if:

\begin{aligned} &(i)\\ &4^{2 x}=\frac{1}{32} \end{aligned} \begin{aligned} &\text { (ii) }\\ &\sqrt{2^{x+3}}=16 \end{aligned}\begin{aligned} &\text { (iii) }\\ &\left(\sqrt{\frac{3}{5}}\right)^{x+1}=\frac{125}{27} \end{aligned}\begin{aligned} &(\text { iv })\\ &\left(\sqrt[3]{\frac{2}{3}}\right)^{x-1}=\frac{27}{8} \end{aligned}

Solve:

\begin{aligned} &\text { (i) }\\ &4^{x-2}-2^{x+1}=0 \end{aligned}\begin{aligned} &(\text { ii })\\ &3^{x^{2}}: 3^{x}=9: 1 \end{aligned}

Solve:

\text { (i) } 8 \times 2^{2 x}+4 \times 2^{x+1}=1+2^{x}\text { (ii) } 2^{2 x}+2^{x+2}-4 \times 2^{3}=0\begin{aligned} &\text { (iii) }\\ &(\sqrt{3})^{x-3}=(\sqrt[4]{3})^{x+1} \end{aligned}

Find the values of m and n if:

4^{2 m}=(\sqrt[3]{16})^{-\frac{6}{n}}=(\sqrt{8})^{2}

Solve for x and y, if:

(\sqrt{32})^{x} \div 2^{y+1}=1 \text { and } 8^{y}-16^{4-\frac{x}{2}}=0

Prove that:

\begin{aligned} &\text { (i) }\\ &\left(\frac{x^{a}}{x^{b}}\right)^{a+b-c} \times\left(\frac{x^{b}}{x^{c}}\right)^{b+c-a} \times\left(\frac{x^{c}}{x^{a}}\right)^{c+a-b}=1 \end{aligned}

\begin{aligned} &\text { (ii) }\\ &\frac{x^{a(b-c)}}{x^{b(a-c)}} \div\left(\frac{x^{b}}{x^{a}}\right)^{c}=1 \end{aligned}

\text { If } a^{x}=b, b^{y}=c \text { and } c^{z}=a, \text { prove that } x y z=1

\begin{aligned} &\text { If } a^{x}=b^{y}=c^{2} \text { and } b^{2}=a c, \text { prove that: }\\ &y;=\frac{2 x z}{z+x} \end{aligned}

\begin{aligned} &\text { If } 5^{-p}=4^{-q}=20^{r} ; \text { show that: }\\ &\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0 \end{aligned}

\begin{array}{l} \text { If } m \neq n \text { and }(m+n)^{-1}\left(m^{-1}+n^{-1}\right)=m^{x} n^{y} \\ \text { Show that: } x+y+2=0 \end{array}

\text { If } 5^{x+1}=25^{x-2} ; \text { find the value of: }3^{x-3} \times 2^{3-x}

\text { If } 4^{x+3}=112+8 \times 4^{x} ; \text { find }(18 x)^{3 x}

Solve for x:

Evaluate:

\text { (i) } 9^{\frac{5}{2}}-3 \times 8^{\circ}-\left(\frac{1}{81}\right)^{-\frac{1}{2}}(64)^{\frac{2}{3}}-\sqrt[3]{125}-\frac{1}{2^{-5}}+(27)^{-\frac{2}{3}} \times\left(\frac{25}{9}\right)^{-\frac{1}{2}}\left[\left(-\frac{2}{3}\right)^{-2}\right]^{3} \times\left(\frac{1}{3}\right)^{-4} \times 3^{-1} \times \frac{1}{6}

Express each of the following in logarithmic form:

\begin{aligned} &\text { (i) }\\ &5^{3}=125 \end{aligned}\begin{aligned} &\text { (ii) }\\ &3^{-2}=\frac{1}{9} \end{aligned}

\begin{aligned} &\text { (iii) }\\ &10^{-3}=0.001 \end{aligned}

\begin{aligned} &\text { (iv) }\\ &(81)^{\frac{3}{4}}=27 \end{aligned}

Simplify:

\frac{3 \times 9^{n+1}-9 \times 3^{2 n}}{3 \times 3^{2 n+3}-9^{n+1}}

Solve:

3^{x-1} \times 5^{2 y-3}=225

\begin{aligned} &\text { If }\\ &\left(\frac{a^{-1} b^{2}}{a^{2} b^{-4}}\right)^{7} \div\left(\frac{a^{3} b^{-5}}{a^{-2} b^{3}}\right)^{-5}=a^{x} \cdot b^{y}, \text { find } x+y \end{aligned}

\text { If } 3^{x+1}=9^{x-3}, \text { find the value of } 2^{1+x}

\begin{aligned} &\frac{1}{2 x}+\frac{1}{4 y}+\frac{1}{8 z}=4\\ &\text { If } 2^{x}=4^{y}=8^{z} \text { and } 2 x \text { 4y } 8 z \text { , find the value of } x \text { . } \end{aligned}

\frac{9^{n} \cdot 3^{2} \cdot 3^{n}-(27)^{n}}{\left(3^{m} \cdot 2\right)^{3}}=3^{-3}

If

Show that: m-n=1

Solve for x:

(13)^{\sqrt{x}}=4^{4}-3^{4}-6

\begin{aligned} &\text { If } 3^{4 x}=(81)^{-1} \text {and }(10)^{\frac{1}{4}}=0.0001 \quad \text { find the value of: }\\ &2^{-x} \times 16^{y} \end{aligned}

3\left(2^{x}+1\right)-2^{x+2}+5=0

\begin{aligned} &\text { If }\left(a^{m}\right)^{n}=a^{m} \cdot a^{n}, \text { find the value of: }\\ &m;(n-1)-(n-1) \end{aligned}

\begin{aligned} &\text { If } m=\sqrt[3]{15} \text { and } n=\sqrt[3]{14}, \text { find the value of }\\ &m-n-;\frac{1}{m^{2}+m n+n^{2}} \end{aligned}

\begin{aligned} &\text { Evaluate: }\\ &\frac{2^{n} \times 6^{m+1} \times 10^{m-n} \times 15^{m+n-2}}{4^{m} \times 3^{2 n+n} \times 25^{m-1}} \end{aligned}

\begin{aligned} &\text { Evaluate: }\\ &\left(\frac{x^{a}}{x^{\prime}}\right)^{\frac{1}{q \prime}} \times\left(\frac{x^{\prime}}{x^{p}}\right)^{\frac{1}{1 p}} \times\left(\frac{x^{p}}{x^{a}}\right)^{\frac{1}{\infty}} \end{aligned}

Prove that:

\begin{aligned} &(\mathrm{i})\\ &\frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}=\frac{2 b^{2}}{b^{2}-a^{2}} \end{aligned}\begin{aligned} &\text { (ii) }\\ &\frac{a+b+c}{a^{-1} b^{-1}+b^{-1} c^{-1}+c^{-1} a^{-1}}=a b c \end{aligned}

Evaluate:

\frac{4}{(216)^{-3}}+\frac{1}{(256)^{-\frac{3}{4}}}+\frac{2}{(243)^{\frac{1}{5}}}