Question

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (49x−2y4)−1/2(xy1/2)

Accepted Answer

Step 1: Begin by simplifying the first term, $$(49x^{-2}y^4)^{-1/2}. $$Recall that raising a power to a negative exponent means taking the reciprocal of the base and then applying the positive exponent. Rewrite it as $$(49x^{-2}y^4)^{-1/2} = \frac{1}{(49x^{-2}y^4)^{1/2}}. $$Step 2: Simplify $$(49x^{-2}y^4)^{1/2} by $$applying the square root to each factor inside the parentheses. Use the property $$(a^m b^n)^{1/2} = a^{m/2} b^{n/2}. $$This gives $$(49x^{-2}y^4)^{1/2} = 7x^{-1}y^2$$, since $$\sqrt{49} = 7$$, $$x^{-2/2} = x^{-1}$$, and $$y^{4/2} = y^2. $$Step 3: Substitute the simplified form back into the expression. The first term becomes $$\frac{1}{7x^{-1}y^2}. $$Combine this with the second term $$xy^{1/2}$$, so the expression is now $$\frac{1}{7x^{-1}y^2} \cdot xy^{1/2}. $$Step 4: Simplify the multiplication. Combine the $$x-$$terms using the property $$x^m \cdot x^n = x^{m+n}. $$Here, $$x^{-1} \cdot x = x^{(-1+1)} = x^0 = 1. $$For the $$y-$$terms, use $$y^2 \cdot y^{1/2} = y^{(2+1/2)} = y^{5/2}. $$The expression now simplifies to $$\frac{1}{7} \cdot y^{5/2}. $$Step 5: Write the final simplified expression as $$\frac{y^{5/2}}{7}. $$This is the fully simplified form of the given expression.

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (49x−2y⁴)^−1/2(xy^1/2)

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Key Concepts

Exponents and Negative Exponents

Radicals and Rational Exponents

Combining Like Terms