Question

use the properties of exponents to simplify each expression. write all answers with positive exponents only. (assume variables are nonzero.)
\\(\frac{(4x^{-6}y^{0})^{4}}{(16x^{2}y^{-4})^{2}} = \\)
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question 26
use the properties of exponents to simplify each expression. write all answers with positive exponents only. (assume variables are nonzero.)
\\(\left(\frac{8x^{2}y^{2}}{4x^{4}y^{-2}}\
ight)^{3}\\)

Explanation:

Step1: Simplify numerator (power of product)

$(4x^{-6}y^6)^4 = 4^4 \cdot (x^{-6})^4 \cdot (y^6)^4 = 256x^{-24}y^{24}$

Step2: Simplify denominator (power of product)

$(16x^2y^{-4})^2 = 16^2 \cdot (x^2)^2 \cdot (y^{-4})^2 = 256x^4y^{-8}$

Step3: Divide numerator by denominator

$\frac{256x^{-24}y^{24}}{256x^4y^{-8}} = x^{-24-4}y^{24-(-8)} = x^{-28}y^{32}$

Step4: Convert to positive exponents

$x^{-28}y^{32} = \frac{y^{32}}{x^{28}}$

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Step5: Simplify base of the second expression

$\frac{8x^2y^2}{4x^4y^{-2}} = 2x^{2-4}y^{2-(-2)} = 2x^{-2}y^4$

Step6: Apply the cube power

$(2x^{-2}y^4)^3 = 2^3 \cdot (x^{-2})^3 \cdot (y^4)^3 = 8x^{-6}y^{12}$

Step7: Convert to positive exponents

$8x^{-6}y^{12} = \frac{8y^{12}}{x^6}$

Answer:

1. $\frac{y^{32}}{x^{28}}$
2. $\frac{8y^{12}}{x^6}$