Question

use the law of exponents and simplify. write your answers in positive exponents. ((2a^{-3}b^{5})^{4}(8a^{-5}b^{-6})^{-2}) (\bigcirc) (dfrac{2b^{8}}{64a^{22}}) (\bigcirc) (dfrac{b^{32}}{2a^{22}}) (\bigcirc) (dfrac{b^{17}}{4a^{12}}) (\bigcirc) (dfrac{b^{32}}{4a^{2}})

Explanation:

Step1: Apply power of a product rule

For \((2a^{-3}b^{5})^{4}\), use \((xy)^n = x^n y^n\) and \((x^m)^n = x^{mn}\):
\(2^4(a^{-3})^4(b^5)^4 = 16a^{-12}b^{20}\)

For \((8a^{-5}b^{-6})^{-2}\), use the same rules:
\(8^{-2}(a^{-5})^{-2}(b^{-6})^{-2} = \frac{1}{8^2}a^{10}b^{12} = \frac{1}{64}a^{10}b^{12}\)

Step2: Multiply the two expressions

Multiply \(16a^{-12}b^{20}\) and \(\frac{1}{64}a^{10}b^{12}\):
\(16 \times \frac{1}{64} \cdot a^{-12 + 10} \cdot b^{20 + 12}\)

Simplify coefficients and exponents:
\(\frac{16}{64}a^{-2}b^{32} = \frac{1}{4}a^{-2}b^{32}\)

Step3: Convert to positive exponents

Use \(a^{-n} = \frac{1}{a^n}\), so \(a^{-2} = \frac{1}{a^2}\). Then:
\(\frac{1}{4} \cdot \frac{b^{32}}{a^2} = \frac{b^{32}}{4a^2}\)

Answer:

\(\boldsymbol{\frac{b^{32}}{4a^2}}\) (corresponding to the last option)