Exercise 1.5 Practice
Laws of Exponents for Real Numbers
Q1: Find Value
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Find:
(i) $81^{\frac{1}{2}}$
(ii) $32^{\frac{1}{5}}$
(iii) $125^{\frac{1}{3}}$
(i) $81^{\frac{1}{2}}$
(ii) $32^{\frac{1}{5}}$
(iii) $125^{\frac{1}{3}}$
(i) $81^{\frac{1}{2}}$:
$81 = 9^2$.
$(9^2)^{\frac{1}{2}} = 9^{2 \times \frac{1}{2}} = 9^1 = 9$.
$81 = 9^2$.
$(9^2)^{\frac{1}{2}} = 9^{2 \times \frac{1}{2}} = 9^1 = 9$.
(ii) $32^{\frac{1}{5}}$:
$32 = 2^5$.
$(2^5)^{\frac{1}{5}} = 2^{5 \times \frac{1}{5}} = 2^1 = 2$.
$32 = 2^5$.
$(2^5)^{\frac{1}{5}} = 2^{5 \times \frac{1}{5}} = 2^1 = 2$.
(iii) $125^{\frac{1}{3}}$:
$125 = 5^3$.
$(5^3)^{\frac{1}{3}} = 5^{3 \times \frac{1}{3}} = 5^1 = 5$.
$125 = 5^3$.
$(5^3)^{\frac{1}{3}} = 5^{3 \times \frac{1}{3}} = 5^1 = 5$.
(i) 9, (ii) 2, (iii) 5
Q2: Evaluate
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Find:
(i) $4^{\frac{3}{2}}$
(ii) $32^{\frac{2}{5}}$
(iii) $16^{\frac{3}{4}}$
(iv) $125^{\frac{-1}{3}}$
(i) $4^{\frac{3}{2}}$
(ii) $32^{\frac{2}{5}}$
(iii) $16^{\frac{3}{4}}$
(iv) $125^{\frac{-1}{3}}$
(i) $4^{\frac{3}{2}}$:
$(2^2)^{\frac{3}{2}} = 2^{2 \times \frac{3}{2}} = 2^3 = 8$.
$(2^2)^{\frac{3}{2}} = 2^{2 \times \frac{3}{2}} = 2^3 = 8$.
(ii) $32^{\frac{2}{5}}$:
$(2^5)^{\frac{2}{5}} = 2^{5 \times \frac{2}{5}} = 2^2 = 4$.
$(2^5)^{\frac{2}{5}} = 2^{5 \times \frac{2}{5}} = 2^2 = 4$.
(iii) $16^{\frac{3}{4}}$:
$(2^4)^{\frac{3}{4}} = 2^{4 \times \frac{3}{4}} = 2^3 = 8$.
$(2^4)^{\frac{3}{4}} = 2^{4 \times \frac{3}{4}} = 2^3 = 8$.
(iv) $125^{\frac{-1}{3}}$:
$(5^3)^{\frac{-1}{3}} = 5^{3 \times \frac{-1}{3}} = 5^{-1} = \frac{1}{5}$.
$(5^3)^{\frac{-1}{3}} = 5^{3 \times \frac{-1}{3}} = 5^{-1} = \frac{1}{5}$.
(i) 8, (ii) 4, (iii) 8, (iv) 1/5
Q3: Simplify
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Simplify:
(i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$
(ii) $(3^{\frac{1}{5}})^4$
(iii) $\frac{7^{\frac{1}{2}}}{7^{\frac{1}{4}}}$
(iv) $13^{\frac{1}{2}} \cdot 17^{\frac{1}{2}}$
(i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$
(ii) $(3^{\frac{1}{5}})^4$
(iii) $\frac{7^{\frac{1}{2}}}{7^{\frac{1}{4}}}$
(iv) $13^{\frac{1}{2}} \cdot 17^{\frac{1}{2}}$
(i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$:
$a^m \cdot a^n = a^{m+n}$.
$2^{\frac{2}{3} + \frac{1}{5}} = 2^{\frac{10+3}{15}} = 2^{\frac{13}{15}}$.
$a^m \cdot a^n = a^{m+n}$.
$2^{\frac{2}{3} + \frac{1}{5}} = 2^{\frac{10+3}{15}} = 2^{\frac{13}{15}}$.
(ii) $(3^{\frac{1}{5}})^4$:
$(a^m)^n = a^{mn}$.
$3^{\frac{1}{5} \times 4} = 3^{\frac{4}{5}}$.
$(a^m)^n = a^{mn}$.
$3^{\frac{1}{5} \times 4} = 3^{\frac{4}{5}}$.
(iii) $\frac{7^{\frac{1}{2}}}{7^{\frac{1}{4}}}$:
$\frac{a^m}{a^n} = a^{m-n}$.
$7^{\frac{1}{2} - \frac{1}{4}} = 7^{\frac{2-1}{4}} = 7^{\frac{1}{4}}$.
$\frac{a^m}{a^n} = a^{m-n}$.
$7^{\frac{1}{2} - \frac{1}{4}} = 7^{\frac{2-1}{4}} = 7^{\frac{1}{4}}$.
(iv) $13^{\frac{1}{2}} \cdot 17^{\frac{1}{2}}$:
$a^m \cdot b^m = (ab)^m$.
$(13 \times 17)^{\frac{1}{2}} = 221^{\frac{1}{2}}$.
$a^m \cdot b^m = (ab)^m$.
$(13 \times 17)^{\frac{1}{2}} = 221^{\frac{1}{2}}$.
(i) $2^{\frac{13}{15}}$, (ii) $3^{\frac{4}{5}}$, (iii) $7^{\frac{1}{4}}$, (iv) $221^{\frac{1}{2}}$