Question

2
rewrite (\frac{1}{1024^{-3/5}}) with positive exponents, and completely simplify, if possible.
select the correct choice below and fill in any answer boxes in your choice.

a. (\frac{1}{1024^{-3/5}} = square) (simplify your answer. type an integer or a fraction.)

b. the solution is not a real number

Explanation:

Step1: Recall negative exponent rule

The rule for negative exponents is \(a^{-n}=\frac{1}{a^{n}}\), so \(\frac{1}{a^{-n}} = a^{n}\). Here, \(a = 1024\) and \(n=\frac{3}{5}\), so \(\frac{1}{1024^{-3/5}}=1024^{3/5}\).

Step2: Rewrite 1024 as a power of 2

We know that \(1024 = 2^{10}\), so substitute \(1024\) with \(2^{10}\) in the expression: \(1024^{3/5}=(2^{10})^{3/5}\).

Step3: Apply exponent power rule

The power rule for exponents is \((a^{m})^{n}=a^{mn}\). So, \((2^{10})^{3/5}=2^{10\times\frac{3}{5}}\).

Step4: Calculate the exponent

Calculate \(10\times\frac{3}{5}\): \(10\times\frac{3}{5}=\frac{30}{5} = 6\). So, \(2^{10\times\frac{3}{5}}=2^{6}\).

Step5: Evaluate \(2^{6}\)

\(2^{6}=64\).

Answer:

A. \(\frac{1}{1024^{-3/5}}=\boxed{64}\)