ICSE Class 8 Maths Selina Solutions Chapter 2 Exponents (Powers)

ICSE Selina Class 8 Maths Solution of Chapter “Exponents (Powers)” is given here. The topics mentioned in the chapter explains about power which means a product of multiplying a number by itself which is usually represented with a base number and an exponent. The topic Laws of exponent with integral powers explains about three laws which need to be kept in mind while solving complex numerical problems. It includes laws of multiplication, division, double exponents, zero exponents, etc.

These detailed solutions will help students to clear all their confusion and learn about Exponents Powers in an easy and understandable way. Students should try solving the questions given in the Selina textbooks and later they can evaluate their answers by comparing with the ICSE Selina Solutions Class 8 Maths Chapter 2 “Exponents (Powers)” solutions provided here at BYJU’S website.

Download ICSE Class 8 Maths Selina Solutions PDF for Chapter 2:-Download Here

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ICSE Class 8 Maths Selina Solutions Chapter 2 Exponents (Powers) – Exercise 2 (A)

Question 1. Evaluate:

(i) \(\left(3^{-1} \times 9^{-1}\right) \div 3^{-2}\)

Solution:

\(=\left(\frac{1}{3} \times \frac{1}{9}\right) \div \frac{1}{3} \times \frac{1}{3}\)

\(=\frac{1}{27}\div \frac{1}{9}\)

(Expressing the equation in fractional form)

\(=\frac{1}{27} \times \frac{9}{1}=\frac{1}{3}\)

(ii) \(\left(3^{-1} \times 4^{-1}\right) \div 6^{-1} \)

Solution:

\(=\left(\frac{1}{3} \times \frac{1}{4}\right) \div \frac{1}{6}\)

\(=\frac{1}{12} \div \frac{1}{6}\)

(Expressing the equation in fractional form)

\(=\frac{1}{12} \times \frac{6}{1}=\frac{1}{2}\)

(iii) \(\left(2^{-1}+3^{-1}\right)^{3}\)

Solution:

\(=\left(\frac{1}{2}+\frac{1}{3}\right)^{3}=\left(\frac{1 \times 3}{2 \times 3}+\frac{1 \times 2}{3 \times 2}\right)^{3}\)

\(=\left(\frac{3+2}{6}\right)^{3}=\left(\frac{5}{6}\right)^{3}\)

(Expressing the equation in fractional form)

\(=\frac{5 \times 5 \times 5}{6 \times 6 \times 6}=\frac{125}{216}\)

(iv) \(\left(3^{-1} \div 4^{-1}\right)^{2} \)

Solution:

\(=\left(\frac{1}{3} \div \frac{1}{4}\right)^{2}\)

(Expressing the equation in fractional form)

\(=\left(\frac{1}{3} \times \frac{4}{1}\right)^{2}=\left(\frac{4}{3}\right)^{2}\)

(Expressing the equation in mixed fraction)

\(=\frac{16}{9}=1 \frac{7}{9}\)

(v) \(\left(2^{2}+3^{2}\right) \times\left(\frac{1}{2}\right)^{2}\)

Solution:

\(=(2 \times 2)+(3 \times 3) \times\left(\frac{1}{2} \times \frac{1}{2}\right) \)

\(=4+9 \times \frac{1}{4}=\frac{13}{4}=3 \frac{1}{4}\) (Simplifying the given equation)

(vi) \(\left(5^{2}-3^{2}\right) \times\left(\frac{2}{3}\right)^{-3}\)

Solution:

\(=(5 \times 5)-(3 \times 3) \times\left(\frac{3}{2}\right)^{3}\)

\(=25-9 \times\left(\frac{3}{2} \times \frac{3}{2} \times \frac{3}{2}\right) \) (Simplifying the given equation)

\(=16 \times \frac{27}{8}=54\)

(vii) \(\left[\left(\frac{1}{4}\right)^{-3}-\left(\frac{1}{3}\right)^{-3}\right]+\left(\frac{1}{6}\right)^{-3} \)

Solution:

\(=\left[\left(\frac{4}{1}\right)^{3}-\left(\frac{3}{1}\right)^{3}\right] \div\left(\frac{6}{1}\right)^{3}\)

\(=\left(\frac{4}{1} \times \frac{4}{1} \times \frac{4}{1}-\frac{3}{1} \times \frac{3}{1} \times \frac{3}{1}\right) \div\left(\frac{6}{1}\right)^{3}\)

\(=64-27 \times\left(\frac{1}{6} \times \frac{1}{6} \times \frac{1}{6}\right) \) (Simplifying the given equation)

\(=37 \times \frac{1}{216}=\frac{37}{216}\)

(viii) \(\left[\left(-\frac{3}{4}\right)^{-2}\right]^{2} \)

Solution:

\(\left[\left(-\frac{3}{4}\right)^{-2}\right]^{2}=\left(-\frac{3}{4}\right)^{-2 \times 2}=\left(-\frac{3}{4}\right)^{-4}\)

\(=\left(\frac{4}{3}\right)^{4}=\frac{4 \times 4 \times 4 \times 4}{3 \times 3 \times 3 \times 3}\)

\(=\frac{256}{81}=3 \frac{13}{81} \)

(Simplifying the given equation)

(ix) \(\left(\left(\frac{3}{5}\right)^{-2}\right)^{-2} \)

Solution:

\(\left\{\left(\frac{3}{5}\right)^{-2}\right\}^{-2}=\left(\frac{3}{5}\right)^{-2 \times (-2)}=\left(\frac{3}{5}\right)^{4}\)

\(=\frac{3 \times 3 \times 3 \times 3}{5 \times 5 \times 5 \times 5}=\frac{81}{625} \) (Simplifying the given equation)

(x) \(\left(5^{-1} \times 3^{-1}\right)+6^{-1} \)

Solution:

\(=\left(\frac{1}{5} \times \frac{1}{3}\right)+\frac{1}{6}\)

\( =\frac{1}{15} \div \frac{1}{6}\) (Simplifying the given equation)

\(=\frac{1}{15} \times \frac{6}{1}=\frac{2}{5}\)

Question 2. \(1125=3^{m} \times 5^{n}\); find m and n

Solution:

\(1125=3^{2} \times 5^{3}\)

The factors of 1125 are 3×3×5×5×5
ICSE Class 8 Maths Selina Solutions Chapter 2 Exponents (Powers) Exercise 2 (A) Question 2

therefore \(1125=3 \times 3 \times 5 \times 5 \times 5\)

Now comparing, \(3^{2} \times 5^{3}=3^{m} \times 5^{n} \)

therefore m=2, n=3

Question 3. Find x, if \( 9 \times 3^{x}=(27)^{2 x-3} \)

Solution:

\(9 \times 3^{x}=(27)^{2 x-3}\)

\(3^{2} \times 3^{x}=(3 \times 3 \times 3)^{2 x-3} \)

(Simplifying the given equation)

\(\Rightarrow 3^{x+2}=(3)^{3(2 x-3)} \)

\(\Rightarrow 3^{x+2}=(3)^{6 x-9}\)

Since bases are same, compare them,
x+2=6x-9
6x-x=9+2

\(\Rightarrow 5 x=11\)

\(\Rightarrow x=\frac{11}{2} \)

(Shifting the terms)

\(\Rightarrow x=2 \frac{1}{5}\)

ICSE Class 8 Maths Selina Solutions Chapter 2 Exponents (Powers) – Exercise 2(B)

Question 1. Compute:

(i) \(1^{8} \times 3^{0} \times 5^{3} \times 2^{2} \)

Solution:

\( 1^{8} \times 3^{0} \times 5^{3} \times 2^{2} \)

\( {=1 \times 1 \times 5 \times 5 \times 5 \times 2 \times 2}\)

\(=125 \times 4 \) (Simplifying the given equation)

=500

(ii) \(\left(4^{7}\right)^{2} \times\left(4^{-3}\right)^{4} \)

Solution:

\( \left(4^{7}\right)^{2} \times\left(4^{-3}\right)^{4} \)

\(=4^{14} \times 4^{-12}\)

\(=4^{14-12}=4^{2} \) (Simplifying the given equation)

\(=4 \times 4=16\)

(iii) \(\left(2^{-9} \div 2^{-11}\right)^{3} \)

Solution:

\(=\left(2^{-9+11}\right)^{3}\)

\(=\left(2^{2}\right)^{3}=2^{6} \)

(Simplifying the given equation)

\(=2 \times 2 \times 2 \times 2 \times 2 \times 2=64\)

(iv) \(\left(\frac{2}{3}\right)^{-4} \times\left(\frac{27}{8}\right)^{-2} \)

Solution:

\(\left(\frac{2}{3}\right)^{-4} \times\left(\frac{27}{8}\right)^{-2}=\left(\frac{2}{3}\right)^{-4} \times\left(\frac{3^{3}}{2^{8}}\right)^{-2}\)

\(=\frac{2^{-4}}{3^{-4}} \times \frac{3^{-6}}{2^{-6}}=\frac{2^{-4}}{2^{-6}} \times \frac{3^{-6}}{3^{-4}}\)

\(=2^{-4+6} \times \frac{1}{3^{-4+6}}=\frac{2^{2}}{3^{2}}=\frac{4}{9}\)

(v) \(\left(\frac{56}{28}\right)^{0} \div\left(\frac{2}{5}\right)^{3} \times \frac{16}{25} \)

Solution:

\(\left(\frac{56}{28}\right)^{0} \div\left(\frac{2}{5}\right)^{3} \times \frac{16}{25}\)

\(=1 \div \frac{2^{3}}{5^{3}} \times \frac{2 \times 2 \times 2 \times 2}{5 \times 5}\)

\(\left[\left(\frac{56}{28}\right)^{0}=1\right] \)

\(=1 \times \frac{5^{3}}{2^{3}} \times \frac{2^{4}}{5^{2}}=5^{3-2} \times 2^{4-3}\)

\(=5^{1} \times 2^{1}=10\)

(vi) \( (12)^{-2} \times 3^{3}\)

Solution:

\(=(2 \times 2 \times 3)^{-2} \times 3^{3}\)

\(=\left(2^{2} \times 3\right)^{-2} \times 3^{3}\)

\(=2^{-2 \times 2} \times 3^{-2} \times 3^{3}\)

\(=2^{-4} \times 3^{-2+3} \times 3^{3}\)

\(=2^{-4} \times 3^{1}\)

\(=\frac{3}{2^{4}}=\frac{3}{2 \times 2 \times 2 \times 2}=\frac{3}{16}\)

(vii) \( (-5)^{4} \times(-5)^{6} \div(-5)^{9}\)

Solution:

\(=(-5)^{4} \times(-5)^{6} \times \frac{1}{(-5)^{9}}\)

\(=(-5)^{4+6-9}\)

\(=(-5)^{1}=-5\)

(viii) \(\left(-\frac{1}{3}\right)^{4} \div\left(-\frac{1}{3}\right)^{8} \times\left(-\frac{1}{3}\right)^{5}\)

Solution:

\(=\left(-\frac{1}{3}\right)^{4} \times \frac{1}{\left(-\frac{1}{3}\right)^{8}} \times\left(-\frac{1}{3}\right)^{5}\)

\(=\left(-\frac{1}{3}\right)^{4+5-8}=\left(-\frac{1}{3}\right)^{9-8}\)

\(=-\frac{1}{3}\)

(ix) \(9^{0} \times 4^{-1} \div 2^{-4}\)

Solution:

\(9^{0} \times 4^{-1} \div 2^{-4}=1 \times \frac{1}{4^{1}} \times \frac{1}{2^{-4}}\)

\(=1 \times \frac{1}{4} \times 2^{4}=1 \times \frac{1}{2^{2}} \times 2^{4}\)

\(=2^{4-2}=2^{2}=4\)

(x) \( (625)^{-\frac{3}{4}} \)

Solution:

\( (625)^{-\frac{3}{4}}=(5 \times 5 \times 5 \times 5)^{-\frac{3}{4}}\)

\(=\left(5^{4}\right)^{-\frac{3}{4}}=5^{4 \times-\frac{3}{4}}\)

\(=5^{-3}=\frac{1}{5^{3}}\)

\(=\frac{1}{5 \times 5 \times 5}\)

\(=\frac{1}{125}\)

(xi) \(\left(\frac{27}{64}\right)^{-\frac{2}{3}} \)

Solution:

\(\left(\frac{27}{64}\right)^{-\frac{2}{3}}=\left[\frac{\left(3^{3}\right)}{\left(4^{3}\right)}\right]^{-\frac{2}{3}}\)

\(=\frac{3^{3 \times-\frac{2}{3}}}{4^{3 \times-\frac{2}{3}}}=\frac{3^{-2}}{4^{-2}}\)

\(=\frac{4^{2}}{3^{2}}=\frac{4 \times 4}{3 \times 3}=\frac{16}{9}=1 \frac{7}{9}\)

(xii) \(\left(\frac{1}{32}\right)^{-\frac{2}{5}}\)

Solution:

\(\left(\frac{1}{32}\right)^{-\frac{2}{5}}=\left(\frac{1}{2 \times 2 \times 2 \times 2 \times 2}\right)^{\frac{2}{5}}\)

\(=\left(\frac{1}{2^{5}}\right)^{-\frac{2}{5}}=\frac{1}{2^{5 \times -\frac{2}{5}}}\)

\(=\frac{1}{2^{-2}}=2^{2}=4\)

(xiii) \( (125)^{-\frac{2}{3}} \div(8)^{\frac{2}{3}}\)

Solution:

\( (125)^{-\frac{2}{8}} \div(8)^{\frac{2}{3}}=\left(5^{3}\right)^{-\frac{2}{3}} \div\left(2^{3}\right)^{\frac{2}{3}}\)

\(=5^{-\frac{2}{3} \times 3} \div 2^{3 \times \frac{2}{3}}\)

\(=5^{-2} \div 2^{2}=\frac{1}{5^{2}} \times \frac{1}{2^{2}}\)

\(=\frac{1}{25} \times \frac{1}{4}=\frac{1}{100}\)

(xiv) \( (243)^{\frac{2}{5}} \div(32)^{-\frac{2}{5}} \)

Solution:

\( =(3 \times 3 \times 3 \times 3 \times 3)^{\frac{2}{5}} \div(2 \times 2 \times 2 \times 2 \times 2)^{-\frac{2}{5}} \)

\(=\left(3^{5}\right)^{\frac{2}{5}} \div\left(2^{5}\right)^{-\frac{2}{5}}\)

\(=3^{5 \times \frac{2}{5}} \div 2^{-\frac{2}{5} \times 5}=3^{2} \div 2^{-2}\)

\(=3^{2} \times \frac{1}{2^{-2}}=3^{2} \times 2^{+2}\)

\(=3 \times 3 \times 2 \times 2=36\)

(xv) \( (-3)^{4}-(\sqrt[4]{3})^{0} \times(-2)^{5} \div(64)^{\frac{2}{3}}\)

Solution:

\(=(-3 \times-3 \times-3 \times-3) -1 \times-2 \times-2 \times-2 \times-2 \times-2 \div\left(2^{6}\right)^{\frac{2}{3}}\)

Note: \( (\sqrt[4]{3})^{0}=1\)

\(=3^{4}+2^{5} \div 2^{6 \times \frac{2}{3}}\)

\(=3^{4}+2^{5} \div 2^{4}=3^{4}+\frac{2^{5}}{2^{4}}\)

\(=3^{4}+2^{5-4}=3^{4}+2=3 \times 3 \times 3 \times 3+2\)

=81+2=83

(xvi) \( (27)^{\frac{2}{3}} \div\left(\frac{81}{16}\right)^{-\frac{1}{4}}\)

Solution:

\( (27)^{\frac{2}{3}} \div\left(\frac{81}{16}\right)^{-\frac{1}{4}}=\left(3^{3}\right)^{\frac{2}{3}} \div\left(\frac{3^{4}}{2^{4}}\right)^{-\frac{1}{4}} \)

\( =3^{3 \times \frac{2}{3}} \div \frac{3^{-\frac{1}{4} \times 4}}{2^{-\frac{1}{4} \times 4}}=3^{2} \div \frac{3^{-1}}{2^{-1}} \)

\(=3^{2} \times \frac{2^{-1}}{3^{-1}}\)

\(=3^{2+1} \times 2^{-1}=3^{3} \times \frac{1}{2^{+1}}\)

\(=\frac{3 \times 3 \times 3}{2}=\frac{27}{2}=13 \frac{1}{2}\)

Question 2. Simplify:

(i) \( 8^{\frac{4}{3}}+25^{\frac{3}{2}}-\left(\frac{1}{27}\right)^{-\frac{2}{3}} \)

Solution:

\(=\left(2^{3}\right)^{\frac{4}{3}}+\left(5^{2}\right)^{\frac{3}{2}}-\left(\frac{1}{3^{3}}\right)^{-\frac{2}{3}}\)

\(=2^{3 \times \frac{4}{3}}+5^{2 \times \frac{3}{2}}-\frac{1}{3^{3}\times\left(\frac{-2}{3}\right)} \)

\(=2^{4}+5^{3}-\frac{1}{3^{-2}}\)

\(=16+125-3^{2}\)

=141-9=132

(ii) \( (64)^{-2} ]^{-3} \div\left[\left\{(-8)^{2}\right\}^{3}\right]^{2} \)

Solution:

\(=\left(2^{6}\right)^{-2 \times-3} \div(-8)^{2 \times 3 \times 2}\)

\(=2^{6 \times(6)} \div(-8)^{12}\)

\(=2^{+36} \div(-8)^{12} \)

\(=2^{+36} \div\left[(-2)^{3}\right]^{12}=2^{36} \div(-2)^{36}\)

\(=\frac{2^{36}}{(-2)^{36}}=\frac{2^{36}}{2^{36}}\) \( (\ 36 \text { is even })\)

\(=2^{36-36}=2^{0}=1 \) (therefore \(a^{0}=1\))

(iii) \(\left(2^{-3}-2^{-4}\right)\left(2^{-3}+2^{-4}\right) \)

Solution:

\(=\left(2^{-3}\right)^{2}-\left(2^{-4}\right)^{2}\)

\(\left\{(a-b)(a+b)=a^{2}-b^{2}\right\}\)

\(=2^{-6}-2^{-8}=\frac{1}{2^{6}}-\frac{1}{2^{8}}\)

\(=\frac{1}{64}-\frac{1}{256}\)

\(=\frac{4-1}{256}=\frac{3}{256}\)

Question 3. Evaluate:

(i) \( (-5)^{0} \)

Solution:

\( (-5)^{0}=1\left( a^{0}=1\right) \)

(ii) \(8^{0}+4^{0}+2^{0} \)

Solution:

\(8^{0}+4^{0}+2^{0}=1+1+1=3\) \( ( a^{0}=1) \)

(iii) \( (8+4+2)^{0} \)

Solution:

\( (8+4+2)^{0}=(14)^{0}=1 \) \(( a^{0}=1) \)

(iv) \(4x^{0} \)

Solution:

\(4x^{0}=4 \times 1=4\)

(v) \( (4x)^{0} \)

Solution:

\( (4x)^{0}=1\)

(vi) \(\left[\left(10^{3}\right)^{0}\right]^{5}\)

Solution:

\(\left[\left(10^{3}\right)^{0}\right]^{5}=10^{3 \times 0 \times 5}=10^{0}=1\)

(vii) \(\left(7x^{0}\right)^{2} \)

Solution:

\(\left(7x^{0}\right)^{2}=7^{2} \times x^{0 \times 2}=49 \times 1=49\)

(viii) \(9^{0}+9^{-1}-9^{-2}+9^{\frac{1}{2}}-9^{-\frac{1}{2}} \)

Solution:

\(9^{0}+9^{-1}-9^{-2}+\frac{1}{9^{\frac{1}{2}}}-9^{-\frac{1}{2}}\)

\( =1+\frac{1}{9}-\frac{1}{9^{2}}+\left(3^{2}\right)^{\frac{1}{2}}-\left(3^{2}\right)^{-\frac{1}{2}} \)

\(=1+\frac{1}{9}-\frac{1}{81}+3^{2 \times \frac{1}{2}}-3^{2 \times\left(-\frac{1}{2}\right)} \)

\(=1+\frac{1}{9}-\frac{1}{81}+3-3^{-1}\)

\(=1+\frac{1}{9}-\frac{1}{81}+\frac{3}{1}-\frac{1}{3}\)

\(=\frac{81+9-1+243-27}{81}=\frac{333-28}{81}\)

\(=\frac{305}{81}=3 \frac{62}{81}\)

Question 4. Simplify:

(i) \(\frac{a^{5} b^{2}}{a^{2} b^{-3}} \)

Solution:

\(\frac{a^{5} b^{2}}{a^{2} b^{-3}}=a^{5-2} \cdot b^{2+3}=a^{3} b^{5}\)

(ii) \(15 y^{8} \div 3 y^{3} \)

Solution:

\(15 y^{8} \div 3 y^{3}=\frac{15 y^{8}}{3 y^{3}}\)

\(=5 y^{\{8-3\}}\)

\(=5 y^{5}\)

(iii) \(x^{10} y^{6} \div x^{3} y^{-2} \)

Solution:

\(x^{10} y^{6} \div x^{3} y^{-2}=\frac{x^{10} y^{6}}{x^{3} y^{-2}}\)

\(=x^{10-3} \cdot y^{6+2}\)

\(=x^{7} y^{8}\)

(iv) \(5z^{16} \div 15z^{-11}\)

Solution:

\(5z^{16} \div 15z^{-11}=\frac{5z^{16}}{15z^{-11}}\)

\(=\frac{5}{15} z^{16+11}\)

\(=\frac{1}{3} z^{27}\)

(v) \(\left(36x^{2}\right)^{\frac{1}{2}} \)

Solution:

\(\left(36x^{2}\right)^{\frac{1}{2}}=(36)^{\frac{1}{2}} \cdot x^{2 \times \frac{1}{2}}\)

\(=(6 \times 6)^{\frac{1}{2}} \cdot x=\left(6^{2}\right)^{\frac{1}{2}} \cdot x\) = 6x

(vi) \(\left(125x^{-3}\right)^{\frac{1}{3}}\)

Solution:

\( \left(125 x^{-3}\right)^{\frac{1}{3}}=(125)^{\frac{1}{3}} x^{-3 \times \frac{1}{3}} \)

\(=(5 \times 5 \times 5)^{\frac{1}{3}} x^{-1}\)

\(\left(5^{3}\right)^{\frac{1}{3}} \cdot x^{-1}=5 x^{-1}\)

\(=\frac{5}{x}=5 x^{-1}\)

(vii) \(\left(2x^{2} y^{-3}\right)^{-2}\)

Solution:

\(\left(2x^{2} y^{-3}\right)^{-2}=2^{-2} x^{2 \times-2} \cdot y^{-3x-2}\)

\(=\frac{1}{2^{2}} x^{-4} \cdot y^{6}\)

\(=\frac{1}{4} \times \frac{y^{6}}{x^{4}}\)

\(=\frac{y^{6}}{4x^{4}}=\frac{1}{4} \cdot y^{6} x^{-4}\)

(viii) \(\left(27 x^{-3} y^{6}\right)^{\frac{2}{3}} \)

Solution:

\(\left(27 x^{-3} y^{6}\right)^{\frac{2}{3}}=(27)^{\frac{2}{3}} \cdot x^{-3 \times \frac{2}{3}} \cdot y^{6 \times \frac{2}{3}}\)

\(=(3 \times 3 \times 3)^{\frac{2}{3}} x^{-2} \cdot y^{4}\)

\(=\left[(3 \times 3 \times 3)^{\frac{1}{3}}\right]^{2} x^{-2} \cdot y^{4}\)

\(=3^{2} x^{-2} y^{4}\)

\(=9x^{-2} y^{4}\)

\(=\frac{9 y^{4}}{x^{2}}=9x^{-2} y^{4}\)

(ix) \(\left(-2 x^{\frac{2}{3}} y^{-\frac{3}{2}}\right)^{6} \)

Solution:

\( =(-2)^{6} x^{\frac{2}{3} \times 6} y^{-\frac{3}{2} \times 6} \)

\(=64x^{4} y^{-9}\)

\(=\frac{64x^{4}}{y^{9}}\)

\(=64 x^{4} y^{-9}\)

Question 5. Simplify:

\(\left(x^{a+b}\right)^{a-b} \cdot\left(x^{b+c}\right)^{b-c} \cdot\left(x^{c+a}\right)^{c-a}\)

Solution:

\(\left(x^{a+b}\right)^{a-b} \cdot\left(x^{b+c}\right)^{b-c} \cdot\left(x^{c+a}\right)^{c-a}\)

\(=x^{(a+b)(a-b)} x^{(b+c)(b-c)} x^{(c+a)(c-a)} \)

\(=x^{a^{2}-b^{2}} x^{b^{2}-c^{2}} x^{c^{2}-a^{2}}\)

\(=x^{a^{2}-b^{2}+b^{2}-c^{2}+c^{2}-a^{2}}\)

\(=x^{0}\)

=1

Question 6.
Simplify:

(i) \(\sqrt[5]{x^{20} y^{-10} z^{5}}\div \frac{x^{3}}{y^{3}} \)

Solution:

\(\sqrt[5]{x^{20} y^{-10} z^{5}} \div \frac{x^{3}}{y^{3}}\)

\(=\left(x^{20} y^{-10} z^{5}\right)^{\frac{1}{5}} \div \frac{x^{3}}{y^{3}}\)

\(x^{20 \times \frac{1}{5}} \cdot y^{-10 \times \frac{1}{5}} \cdot z^{5 \times \frac{1}{5}} \div \frac{x^{3}}{y^{3}}\)

\(=x^{4} \cdot y^{-2} \cdot z^{1} \times \frac{y^{3}}{x^{3}}\)

\(=x^{4-3} \cdot y^{-2+3} \cdot z^{1} \)

= xyz

(ii) \(\left(\frac{256 a^{16}}{81b^{4}}\right)^{\frac{-3}{4}}\)

Solution:

\(\left[\frac{256 a^{16}}{81 b^{4}}\right]^{-\frac{3}{4}}=\left[\frac{4^{4} a^{16}}{3^{4} b^{4}}\right]^{\frac{-3}{4}}\)

Where \(256=4 \times 4 \times 4 \times 4=4^{4}\)

\(81=3 \times 3 \times 3 \times 3=3^{4}\)

\(=\frac{4^{4 \times \frac{-3}{4}} \cdot a^{16 \times \frac{-3}{4}}}{3^{4 \times \frac{-3}{4}} b^{4 \times \frac{-3}{4}}}\)

\(=\frac{4^{-3} \cdot a^{-12}}{3^{-3} \cdot b^{-8}} \)

\(=\frac{3^{3} b^{3}}{4^{3} a^{12}}\)

\(=\frac{27 b^{3}}{64 a^{12}}\)

\(=\frac{27}{64} \cdot a^{-12} b^{3}\)

Question 7

{i} \(\left(a^{-2}\right)^{-2} \cdot(ab)^{-3}\)

Solution:

\(\left(a^{-2}\right)^{-2} \cdot(ab)^{-3}\)

\(=\left(a^{-2 \times -2} \cdot b^{-2}\right) \cdot\left(a^{-3} \cdot b^{-3}\right) \)

\(=a^{+4} \cdot b^{-2} \cdot a^{-3} \cdot b^{-3}\)

\(=a^{4-3} \cdot b^{-2-3} \)

\(=a b^{-5}\)

\(=\frac{a}{b^{5}}\)

(ii) \(\left(x^{n} y^{-m}\right)^{4} \times\left(x^{3} y^{-2}\right)^{-n}\)

Solution:

\(\left(x^{n} y^{-m}\right)^{4} \times\left(x^{3} y^{-2}\right)^{-n} =x^{4n} y^{-4m} \times x^{-3n} y^{2n} \)

\(=x^{4n-3n} \cdot y^{-4m+2n}\)

\(=x^{n} y^{-4m+2n}\)

(iii) \(\left(\frac{125 a^{-3}}{y^{6}}\right)^{\frac{-1}{3}}\)

Solution:

\(\left[\frac{125 a^{-3}}{y^{6}}\right]^{ \frac{-1}{3}}=\left[\frac{5^{3} a^{-3}}{y^{6}}\right]^{ \frac{-1}{3}}\)

Where \(125=5 \times 5 \times 5=5^{3}\)

\(=\frac{5^{3 \times \frac{-1}{3}} \cdot a^{-3 \times \frac{-1}{3}}}{y^{6 \times \frac{-1}{3}}}\)

\(=\frac{5^{-1} \cdot a^{1}}{y^{-2}}\)

\( =\frac{a \cdot y^{2}}{5} \)

(iv) \(\left(\frac{32 x^{-5}}{243 y^{-5}}\right)^{\frac{-1}{5}}\)

Solution:

\(\left[\frac{32 x^{-5}}{243 y^{-5}}\right]^{\frac{-1}{5}}=\left[\frac{2^{5} x^{-5}}{3^{5} y^{-5}}\right]^{\frac{-1}{5}}\)

Where \(32=2 \times 2 \times 2 \times 2 \times 2=2^{5} \)

\(243=3 \times 3 \times 3 \times 3 \times 3=3^{5}\)

\(=\frac{2^{5 \times \frac{-1}{5}} \cdot x^{-5 \times \frac{-1}{5}}}{3^{5 \times \frac{-1}{5}} y^{-5 \times \frac{-1}{5}}}\)

\(=\frac{2^{-1} x^{+1}}{3^{-1} y^{+1}}\)

\(=\frac{3x}{2y}\)

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

Solution:

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

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

\(=a^{-1} b^{\frac{1}{2}} \times a^{\frac{1}{3}} b^{-1}\)

\(=a^{-1+\frac{1}{3}} b^{\frac{1}{2}-1}\)

\(=a^{-\frac{2}{3}} b^{-\frac{1}{2}}\)

\(=\frac{1}{a^{\frac{2}{3}} b^{\frac{1}{2}}}\)

(vi) \( (xy)^{m-n} \cdot(yz)^{n-l} \cdot(zx)^{I-m}\)

Solution:

\( (xy)^{m-n} \cdot (yz)^{n-l} \cdot (x z)^{l-m}\)

\(=x^{m-n} \cdot y^{m-n} \cdot y^{n-l} \cdot z^{n-l} x^{l-m} \cdot z^{l-m}\)

\(=x^{m-n+l-m} \cdot y^{m-n+n-l} \cdot z^{n-l+l-m}\)

\(=x^{l-n} \cdot y^{m-l} \cdot z^{n-m}\)

Question 8.

Show that:

\(\left(\frac{x^{a}}{x^{-b}}\right)^{a-b} \cdot \left(\frac{x^{b}}{x^{-c}}\right)^{b-c} \cdot \left(\frac{x^{c}}{x^{-a}}\right)^{c-a}=1\)

Solution:

L.H.S. \(=\left(\frac{x^{a}}{x^{-b}}\right)^{a-b} \cdot \left(\frac{x^{b}}{x^{-c}}\right)^{b-c} \cdot \left(\frac{x^{c}}{x^{-a}}\right)^{c-a}\)

\(=\left(x^{a+b}\right)^{a-b} \cdot\left(x^{b+c}\right)^{b-c} \cdot\left(x^{c+a}\right)^{c-a}\)

\(=x^{(a+b)(a-b)} x^{(b+c)(b-c)} x^{(c+a)(c-a)} \)

\(=x^{a^{2}-b^{2}} x^{b^{2}-c^{2}} x^{c^{2}-a^{2}}\)

\(=x^{a^{2}-b^{2}+b^{2}-c^{2}+c^{2}-a^{2}}\)

\(=x^{0}\)

=1 = R.H.S

Question 9.

Evaluate:

\(\frac{x^{5+n}\left(x^{2}\right)^{3n+1}}{x^{7n-2}}\)

Solution:

\(\frac{x^{5+n} \left(x^{2}\right)^{3n+1}}{x^{7n-2}}\)

\(=\frac{x^{5+n} \times x^{2(3n+1)}}{x^{7n-2}}\)

\(=\frac{x^{5+n} \times x^{6n+2}}{x^{7n-2}}\)

\(=x^{5+n+6n+2-7n+2}\)

\(=x^{9}\)

Question 10. Evaluate:

\(\frac{a^{2n+1} \times a^{(2n+1)(2n-1)}}{a^{n(4n-1) \times\left(a^{2}\right)^{2n+3}}}\)

Solution:

\(\frac{a^{2n+1} \times a^{(2n+1)(2n-1)}}{a^{n(4n-1) \times\left(a^{2}\right)^{2n+3}}}\)

\(=\frac{a^{2n+1} \times a^{(2n)^{2}-(1)^{2}}}{a^{4n^{2}-n} \times a^{2(2n+3)}} \)

\(=\frac{a^{2n+1} \times a^{4n^{2}-1}}{a^{4n^{2}-n} \times a^{4n+6}}\)

\(=a^{2n+1+4n^{2}-1-4n^{2}+n-4n-6}\)

\(=a^{-n-6}\)

\(=a^{-(n+6)} \)

\(=\frac{1}{a^{n+6}}\)

Question 11.

\( (m+n)^{-1}\left(m^{-1}+n^{-1}\right)=(m n)^{-1} \)

Solution:

L.H.S. \( =(m+n)^{-1}\left(m^{-1}+n^{-1}\right) \)

\(=\frac{1}{m+n}\left(\frac{1}{m}+\frac{1}{n}\right)=\frac{1}{m+n} \cdot \frac{n+m}{m n}=\frac{1}{m n}\)

\(=(m n)^{-1}\)

=R.H.S.
Hence proved.

Question 12. Prove that:

(i) \(\left(\frac{x^{a}}{x^{b}}\right)^{\frac{1}{a b}}\left(\frac{x^{b}}{x^{c}}\right)^{\frac{1}{b c}}\left(\frac{x^{c}}{x^{a}}\right)^{\frac{1}{c a}}=1\)

Solution:

\(\left(\frac{x^{a}}{x^{b}}\right)^{\frac{1}{a b}}\left(\frac{x^{b}}{x^{c}}\right)^{\frac{1}{b c}}\left(\frac{x^{c}}{x^{a}}\right)^{\frac{1}{c a}}=1\)

L.H.S \(=\left(\frac{x^{a}}{x^{b}}\right)^{\frac{1}{a b}}\left(\frac{x^{b}}{x^{c}}\right)^{\frac{1}{b c}}\left(\frac{x^{c}}{x^{a}}\right)^{\frac{1}{c a}}\)

\(=\left(x^{a-b}\right)^{\frac{1}{b b}}\left(x^{b-c}\right)^{\frac{1}{b_{c}}}\left(x^{c-a}\right)^{\frac{1}{m}}\)

\(=x^{\frac{a-b}{a b}} \cdot x^{\frac{b-c}{b c}} \cdot x^{\frac{c-a}{c a}}\)

\(=x^{\frac{a-b}{a b}+\frac{b-c}{b c}+\frac{c-a}{c a}}\)

\(=x^{\frac{a c-b c+a b-a c+b c-a b}{a b c}}\)

\(=x^{0}=1 \)=R.H.S

(ii) \(\frac{1}{1+x^{a-b}}+\frac{1}{1+x^{b-a}}=1\)

Solution:

\(\frac{1}{1+x^{a-b}}+\frac{1}{1+x^{b-a}}=1\)

L.H.S. \(=\frac{1}{1+x^{a-b}}+\frac{1}{1+x^{b-a}}\)

\(=\frac{1}{x^{a-a}+x^{a-b}}+\frac{1}{x^{b-b}+x^{b-a}}\)

\(=\frac{1}{x^{a} x^{-a}+x^{a} x^{-b}}+\frac{1}{x^{b} x^{-b}+ x^{b} x^{-a}}\)

\(=\frac{1}{x^{a}\left(x^{-a}+x^{-b}\right)}+\frac{1}{x^{b}\left(x^{-b}+x^{-a}\right)} \)

\(=\frac{1}{\left(x^{-a}+x^{-b}\right)}\left[\frac{1}{x^{a}}+\frac{1}{x^{b}}\right] \)

\(=\frac{1}{x^{-a}+x^{-b}}\left[x^{-a}+x^{-b}\right] \)

= 1 = R.H.S

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