Prove that:(i) \(\sqrt{3\times5^{-3}}\div \sqrt[3]{{3}^{-1}}\sqrt5\)�

1.	Prove that:(i) \(\sqrt{3\times5^{-3}}\div \sqrt[3]{{3}^{-1}}\sqrt5\) x \(\sqrt[6]{{3}\times5^6}=\frac{3}{5}\)(ii) \({9}^{\frac{3}{2}}-3\times5^0-(\frac{1}{81})^{-\frac{1}{2}}\) = 15(iii) \((\frac{1}{4})^{-2}-3\times8^{\frac{2}{3}}\times4^0+(\frac{9}{15})^{-\frac{1}{2}}\) = \(\frac{16}{3}\)(iv) \(\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{{10^{-\frac{1}{5}}}{}\times{5{\frac{3}{5}}}}\)\(\frac{3^{\frac{4}{3}\times {5}^{-\frac{7}{5}}}}{4^{\frac{3}{5}\times6}}\)
Answer» (i) (3^1/2+1/6.5^-3/2 ⁺¹) / (3^-1/3.5^1/2) =(3 ^2/3.5^-1/2) / (3^-1/3.5^1/2) =(3 ^2/3 + ^1/3) / (5^1/2 ^+1/2) =3/5 (ii) (3 ² )^3/2 -3.1 – (1/9²) ^-1/2 = 3 ³ -3 -9 =27 -3 -9 =27-12 =15 (iii) 2 ^(-2)(-2) -3.8^2/3 +(3/4)^-1 =2 ⁴-3.2² + 4/3 =16 -12 + 4/3 =16/3 (iv) [(2.3 ^1/3)/(2^-1/5 5^2/5)] × (2^-1/5.3)/ (3^4/3.5^7/5) = 2.3 ^1/3 ^{+1 -4/3} / 5^2/5-7/5 = 2.5 =10

Prove that:(i) \(\sqrt{3\times5^{-3}}\div \sqrt[3]{{3}^{-1}}\sqrt5\) x \(\sqrt[6]{{3}\times5^6}=\frac{3}{5}\)(ii) \({9}^{\frac{3}{2}}-3\times5^0-(\frac{1}{81})^{-\frac{1}{2}}\) = 15(iii) \((\frac{1}{4})^{-2}-3\times8^{\frac{2}{3}}\times4^0+(\frac{9}{15})^{-\frac{1}{2}}\) = \(\frac{16}{3}\)(iv) \(\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{{10^{-\frac{1}{5}}}{}\times{5{\frac{3}{5}}}}\)\(\frac{3^{\frac{4}{3}\times {5}^{-\frac{7}{5}}}}{4^{\frac{3}{5}\times6}}\)

Answer»

(i) (3^1/2+1/6.5^-3/2 ⁺¹) / (3^-1/3.5^1/2)

=(3 ^2/3.5^-1/2) / (3^-1/3.5^1/2)

=(3 ^2/3 + ^1/3) / (5^1/2 ^+1/2)

=3/5

(ii) (3 ² )^3/2 -3.1 – (1/9²) ^-1/2

= 3 ³ -3 -9

=27 -3 -9

=27-12

=15

(iii) 2 ^(-2)(-2) -3.8^2/3 +(3/4)^-1

=2 ⁴-3.2² + 4/3 =16 -12 + 4/3

=16/3

(iv) [(2.3 ^1/3)/(2^-1/5 5^2/5)] × (2^-1/5.3)/ (3^4/3.5^7/5)

= 2.3 ^1/3 ^{+1 -4/3} / 5^2/5-7/5

= 2.5

=10