Question 1:
Find:\( \)
(i) \(64^{\dfrac{1}{2}}\)
(ii)\(32^{\dfrac{1}{5}}\)
(iii)\(125^{\dfrac{1}{3}}\)
Solution:
(i)\(64^{\dfrac{1}{2}}\)
=\({(8 \times 8)}^{\dfrac{1}{2}}\)
=\({{8}^2}^{\times \dfrac{1}{2}}\)
=\(8\)
(ii)\(32^{\dfrac{1}{5}}\)
=\({(2 \times 2 \times 2 \times 2 \times 2)}^{\dfrac{1}{5}}\)
=\({{2}^5}^{\times \dfrac{1}{5}}\)
=\(2\)
(iii)\(125^{\dfrac{1}{3}}\)
=\({(5 \times 5 \times 5 )}^{\dfrac{1}{3}}\)
=\({{5}^3}^{\times \dfrac{1}{3}}\)
=\(5\)
Question 2:
Find:
(i)\(9^{\dfrac{3}{2}}\)
(ii)\(32^{\dfrac{2}{5}}\)
(iii)\(16^{\dfrac{3}{4}}\)
(iv)\(125^{\dfrac{-1}{3}}\)
Solution:
(i).\(9^{\dfrac{3}{2}}\)
=\({3 \times 2}^{\dfrac{3}{2}}\)
=\({{3}^2}^{\times \dfrac{3}{2}}\)
=\(3^3=27\)
(ii)\(32^{\dfrac{2}{5}}\)
=\(({2 \times 2 \times 2 \times 2 \times 2})^{\dfrac{2}{5}}\)
=\({{2}^5}^{\times \dfrac{2}{5}}\)
=\(4\)
(iii)\(16^{\dfrac{3}{4}}\)
=\({(2 \times 2 \times 2 \times 2) }^{\dfrac{3}{4}}\)
=\({{2}^4}^{\times \dfrac{3}{4}}\)
=\(2^3=8\)
(iv)\(125^{\dfrac{-1}{3}}\)
=\({(5 \times 5 \times 5) }^{\dfrac{-1}{3}}\)
=\({{5}^3}^{\times \dfrac{-1}{3}}\)
=\(5^{-1}=\dfrac{1}{5}\)
Question 3:
Simplify:
(i) \(2^{\dfrac{2}{3}}.2^{\dfrac{1}{5}}\)
(ii)\(({3^{\dfrac{1}{3}}})^7\)
(iii)\(\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}\)
(iv) \(7^{\dfrac{1}{2}}.8^{\dfrac{1}{2}}\)
Solution 3:
(i) \(2^{\dfrac{2}{3}}.2^{\dfrac{1}{5}}\)
= \(2^{{\dfrac{2}{3}}+{\dfrac{1}{5}}}\)
=\(2^{{\dfrac{10+3}{15}}}\)
=\(2^{{\dfrac{13}{15}}}\)
(ii) \(\Bigg({3^{\dfrac{1}{3}}}\Bigg)^7\)
=\(\Bigg(3\Bigg)^{\dfrac{7}{3}}\)
(iii)\(\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}\)
=\(11^{\dfrac{1}{2}-\dfrac{1}{4}}\)
=\(11^{\dfrac{2-1}{4}}\)
=\(11^{\dfrac{1}{4}}\)
(iv) \(7^{\dfrac{1}{2}}.8^{\dfrac{1}{2}}\)
=\({(7 \times 8)}^{\dfrac{1}{2}}\)
=\({(56)}^{\dfrac{1}{2}}\)
