SChand CLASS 9 Chapter 6 Indices/Exponent Exercise 6(A)

 Exercise 6(A)

Question 1

Write the products in the exponential form:

(i) (x . x . x . x. x) (x. x)

(ii) -1 (n . n. n) (n . n)

(iii) $y^8 \times y^5 \times y^2$

(iv) $– x (- x^4)$

(v) $(3a^7b^8c^9)\times (5a^{27}b^{16}c^8)$

(vi) $(x + 2)^2 . (x + 2)^4$

(vii) $(2m – 32n)^{3a-2b} \times  (2m – 3n)^{6a+10b}$

(viii) $x^{2a+b-c}.x^{2c+a-b}.x^{2b+c-a}$

Sol :

(i) (x.x.x.x.x)(x.x)

$x^{5} \cdot x^{2}$

$=x^{m} \cdot x^{n} \Rightarrow x^{m+n}$

$=\quad x^{5+2} \Rightarrow x^{-2}$

(ii) $-1 .(n \cdot n, n)(n \cdot n)$

$-1 n^{3} \cdot n^{2}$

$\because x^{m} \cdot x^{n}=x^{m+n}$

$-1 n^{3+2}$

$\Rightarrow-n^{5}$

(iii) $y^{8} \times y^{5} \times y^{2}$

$\Rightarrow y^{8+5+2}$

$\Rightarrow y^{15}$

(iv) $-x\left(-x^{5}\right)$

$\Rightarrow x \times x^{5}$

$\Rightarrow x^{1+5}$

$\Rightarrow x^{6}$

(v) $\left(3 a^{7} b^{8} c^{9}\right) \times 5\left(a^{27} b^{16} c^{8}\right)$

$3 \times a^{7} \times b^{8} \times c^{9} \times 5 \times c^{27} \times b^{16} \times c^{8}$

∵$x^{m} \cdot x^{n}=x^{m+n}$

$3 \times 5\left(a^{7} \times a^{27}\right)\left(b^{8} \times b^{16}\right)\left(c^{9} \times c^{8}\right)$

$3 \times 5 a^{7+27} \times b^{8+16} c^{9+8}$

$15 a^{34} b^{24} c^{17}$

(vi) $(2 m+3 n)^{3 n-2 b} \times(2 m-3 n)^{6 a+10 b}$

$\because x^{m} \cdot x^{n} \Rightarrow x^{m+n}$

$(2 m-3 n)^{3 a-2 b+6 a+10 b}$

$(2 m-3 n)^{(3 a+6 a-2 b+10 b)}$

$(2 m-3 n)^{9 a+0 b}$

(vii) $x^{2 a+b-c} \cdot x^{2 c+a-b} \cdot x^{2 b+c-a}$

∵$x^{m} x^{n} \cdot x^{0} \Rightarrow x^{m+n+0}$

$\Rightarrow x^{2 a+b-c+2c+a-b+2b+c-a}$

$\Rightarrow x^{2 a+2c+2 b}$

Question 2

Write each expression in the simpler form:

(i) xy³

(ii) $(-x)^5$

(iii) $(-2xy)^4$

(iv) $\frac{1}{2}$

(v) $(x^2)^5$

(vi) $(7^3)^8$

(vii) $(6a^2)^3$

(viii) $(-x^2y^3)^3$

(ix) $(p^2)^5 \times (p^3)^2$

(x) $3(x^4y^3)^{10} \times 5(x^2y^2)^3$

(xi) $\left(\frac{c^3}{d^2}\right)^7$

(xii) $\left(\frac{3p^2}{4q^2}\right)^n$

(xiii) $\left(\frac{a^2b^2}{x^2y^3}\right)^m$

Sol :

(i) $(xy)^{3}$

∵$\left(xy\right)^{m}=x^{m} \cdot y^{m}$

$(x y)^{3}=x^{3} \cdot y^{3}$

(ii) $(-x)^{5}$

⇒(-x)×(-x)×(-x)×(-x)×(-x)

⇒(-x)⁵

(iii) $(-2 x y)^{4}$

⇒(-2xy)×(-2xy)×(-2xy)×(-2xy)

⇒(-2xy)⁴

⇒(16x⁴y⁴)

(iv) $\left(\frac{p}{q}\right)^{8}$

⇒$\frac{p^{8}}{q^{8}}$

(v) $\left(x^{2}\right)^{5}$

$=x^{2 \times 5}=x^{10}$

(vi) $\left.7^{3}\right)^{8}$

⇒$7^{3 \times 8}$

⇒$7^{24}$

(vii) $\left(6 a^{2}\right)^{3}$

⇒$\left(6 a^{2}\right) \times\left(6a^{2}\right) \times\left(6a^{2}\right)$

⇒$\left(216a^{2+2+2}\right)$

⇒$216a^6$

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

⇒$\left(-x^{2} y^{3}\right) \times\left(-x^{2} y^{3}\right) \times\left(-x^{2} y^{3}\right)$

⇒$\left(-x^{2+2+2} \cdot y^{3+3+3}\right)$

⇒$-x^{6} y^{6}$

(ix) $\left(p^{2}\right)^{5} \times\left(p^{3}\right)^{2}$

⇒$p^{2 \times 5+3 \times 2}$

⇒$p^{10+6}$

⇒$p^{16}$

(x) $3\left(x^{4} y^{3}\right)^{10} \times 5\left(x^{2} y^{2}\right)^{3}$

⇒$3\left(x^{4} y^{10}\cdot y^{3 \times 10}\right) \times 5\left(x^{2 \times 3} \cdot y^{2 \times 3}\right)$

⇒$3 \times 5\left(x^{40} \cdot y^{30} \cdot x^{6} \cdot y^{6}\right)$

⇒$15\left(x^{40+6} y^{30+6}\right)$

⇒$15 \cdot\left(x^{46} y^{36}\right)$

⇒$15 x^{46} y^{30}$

(xi) $\left(\frac{c^{3}}{d^{2}}\right)^{7}$

⇒$\frac{\left(c^{3}\right)^{7}}{\left(d^{2}\right)^{7}}$

⇒$\frac{c^{21}}{d ^{14}}$

(xii) $\left(\frac{3p^2}{4q^2}\right)^a$

$=\frac{3^{a} \cdot p^{2 a}}{4^{a} q^{2a}}$

(xiii) $\left(\frac{a^{2} b^{2}}{x^{2} y^{3}}\right)^{m}$

$=\frac{\left(a^{2} b^{2}\right)^{m}}{\left(x^{2} y^{3}\right)^{m}}$

$= \frac{a^{2 m} \times b^{2 m}}{x^{2 m} \times y^{3 m}}$

Question 3

Find the quotient:

(i) $x^6 + x^2$

(ii) $x^{2a} + x^4$

(iii) $\frac{p^5q^3}{p^3q^2}$

(iv) $(−8x^{27}y^{21})÷(−16x^6y^{17})$

(v) $(−8x^{27}y^{21})÷(−16x^6y^{17})$

(vi) $\frac{4pq^2(−5pq^3)}{10p^2q^2}$

(vii) $\frac{(−4ab^2)^2}{16ab}$

(viii) $\frac{x^{a−b}y^{c−d}}{x^{2b−a}y^c}$

(ix) $\frac{(m^{3n−9})^6}{m^{2n−4}}$

(x) $\left[\frac{(x^{2a−4})^2}{x^{a+5}}\right]^3$

Sol :

(i) $x^{6} \div x^{2}$

∵$x^{m} \div x^{n} \Rightarrow x^{m-n}$

$=x^{6} \div x^{2}$

$=x^{6-2}=x^{4}$

(ii) $x^{2} \div x^a$

$=x^{2a-a}$

$=x^{a}$

(iii) $\frac{p^{5} q^{3}}{p^{3} q^{2}}$

$=p^{5} q^{3} \div p^{3} q^{2}$

$=p^{5-2} q^{3-2}$

$=p^{2} q^1$

$=p^{2} q$

(iv) $\frac{-35 x^{10} y^{5}}{-7 x^{3} y^{3}}$

$=-35 x^{10} y^{5} \div\left(-7 x^{3} y^{3}\right)$

$=\frac{35^{5}}{7} x^{10-3} \cdot y^{5-3}$

$=5 x^{7} \cdot y^{2}$

(v) $-8 x^{27} y^{21} \div\left(-16 x^{6} y^{17}\right)$

$=\frac{-8}{-162} x^{27-6} y^{21-17}$

$=\frac{1}{2} x^{21} y^{4}$

(vi) $\frac{4 p q^{2}\left(-5 p q^{3}\right)}{10 p^{2} q^{2}}$

$=\frac{4 \times -5 .p^{1+1}.q^{2+3}}{10p^2q^2}$

$=\frac{-20p^2q^5}{10p^2q^2}$

$=-2 p^{2-2} q^{5-2}=-2p^0q^2$

$=-2q^2$

(vii) $\frac{\left(-4 a b^{2}\right)^{2}}{16 a b}$

$=\frac{16 a^{2} b^{2+2}}{16 a b}$

$=a^{2}.b^{4} \div a b=a^{2-1}.b^{4-1}$

$=a b^{3}$

(viii) $\frac{x^{a-b} y^{c-d}}{x^{2 b-a} y^{c}}$

$=x^{(a-b)(2 b-a)} y^{c-d-c}$

$=x^{a-b-2 b+a} y^{-d}$

$=x^{2 a-3 b} y^{-d}$

(ix) $\frac{\left(m^{3 n-9}\right)^{6}}{m^{2 n-4}}$

$=\frac{m^{3 n \times 6-9 \times 6}}{m^{2 n-4}}$

$=\frac{m^{18 n-54}}{m^{2 n-4}}$

$=m^{10 n-54-(2 n-4)}$

$=m^{18 n-54-2 n+4}$

$=m^{16n-50}$

(x) $\left[\frac{\left(x^{2a-4}\right)^{2}}{x^{a+5}}\right]^{3}$

$\Rightarrow\left(\frac{x^{2 \times 2 a-4 \times 2}}{x^{9+5}}\right)^{3}$

$\Rightarrow\left(\frac{x^{4 a-8}}{x^{a+5}}\right)^{3}$

$\Rightarrow\left(x^{4 a-8-(a+5)}\right)^{3}$

$\Rightarrow\left(x^{4 a-8-a-5}\right)^{3}$

$\Rightarrow\left(x^{3 a-13}\right)^{3}$

$\Rightarrow 3^{9a-39}$