Question

State whether the given statement is True or False: ${(-\frac{3}{5})}^{100}=\left(\frac{-3^{100}}{-5^{100}}\right)$

• A. True
• B. False

Answer 1

The correct option is A True

Determine the given statement ${(-\frac{3}{5})}^{100}=\left(\frac{-3^{100}}{-5^{100}}\right)$

Evaluate L.H.S :

$\begin{array}{rcl}{(-\frac{3}{5})}^{100}& =& {\left(\frac{-1\times 3}{5}\right)}^{100}\\ & =& \frac{({\left(-1\right)}^{100}\times 3^{100})}{5^{100}}[Since{\mathbf{(}\mathbf{a}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{b}\mathbf{)}}^{\mathbf{m}}\mathbf{}\mathbf{=}\mathbf{}{\mathbf{a}}^{\mathbf{m}}\mathbf{}\mathbf{\times }\mathbf{}{\mathbf{b}}^{\mathbf{m}}]\mathrm{and}[\mathbf{Since}\mathbf{}{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{}\mathbf{if}\mathbf{}\mathbf{n}\mathbf{}\mathbf{is}\mathbf{}\mathbf{even}]\\ & =& \frac{(1^{100}\times 3^{100})}{5^{100}}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{-}{\mathbf{1}}^{\mathbf{100}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{1}}^{\mathbf{100}}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{]}\\ & =& \frac{3^{100}}{5^{100}}\end{array}$

Solve R.H.S part:

$\left(\frac{-3^{100}}{-5^{100}}\right)=\left(\frac{3^{100}}{5^{100}}\right)$

Hence, the L.H.S = R.H.S so, the given statement is “True”.