Question

In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (2^−1x^−3y^−1)^−2(2x^−$$6y^4$$)^−$$2(9x^3y$$^−$$3)^0/(2x$$^−4y^−$$6)^2$$

Accepted Answer

Step 1: Simplify each term in the numerator and denominator separately. Start with the first term $$(2^{-1}x^{-3}y^{-1})^{-2}. $$Apply the power rule $$(a^m)^n = a^{m \cdot n} to $$distribute the $$-2$$ exponent to each base: $$2^{(-1)(-2)}x^{(-3)(-2)}y^{(-1)(-2)} = 2^2x^6y^2. $$Step 2: Simplify the second term $$(2x^{-6}y^4)^{-2}. $$Again, apply the power rule $$(a^m)^n = a^{m \cdot n}$$: $$2^{1 \cdot -2}x^{-6 \cdot -2}y^{4 \cdot -2} = 2^{-2}x^{12}y^{-8}. $$Step 3: Simplify the third term $$(9x^3y^{-3})^0. $$Any expression raised to the power of 0 is equal to 1, so $$(9x^3y^{-3})^0 = 1. $$Step 4: Simplify the denominator $$(2x^{-4}y^{-6})^2. $$Apply the power rule $$(a^m)^n = a^{m \cdot n}$$: $$2^{1 \cdot 2}x^{-4 \cdot 2}y^{-6 \cdot 2} = 2^2x^{-8}y^{-12}. $$Step 5: Combine all the simplified terms into a single expression. The numerator becomes $$2^2x^6y^2 \cdot 2^{-2}x^{12}y^{-8} \cdot 1$$, and the denominator is $$2^2x^{-8}y^{-12}. $$Use the product rule $$a^m \cdot a^n = a^{m+n} to $$combine like bases in the numerator, and then simplify the fraction by subtracting exponents for like bases $$a^m / a^n = a^{m-n}.$$

In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (2^−1x^−3y^−1)^−2(2x^−6y^4)^−2(9x^3y^−3)^0/(2x^−4y^−6)^2

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Key Concepts

Exponential Rules

Negative Exponents

Zero Exponent Rule