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Chapter 0: Problem 111

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(49 x^{-2} y^{4}\right)^{-\frac{1}{2}}\left(x y^{\frac{1}{2}}\right) $$

Short Answer

Expert verified
The simplified form of the expression \((49x^{-2}y^{4})^{-\frac{1}{2}}(xy^{\frac{1}{2}})\) is \(\frac{x^{2}y^{\frac{5}{2}}}{7}\).

Step by step solution

01

Apply Power of Power Rule

Apply the power of a power rule to the expression. This rule states that \((a^{m})^{n} = a^{mn}\). Thus, the expression simplifies to \(49^{-\frac{1}{2}} (x^{-2})^{-\frac{1}{2}} (y^{4})^{-\frac{1}{2}} (x y^{\frac{1}{2}})\).
02

Simplify Fractions Further

Simplify each of the fractions further by multiplying the exponents. \(49^{-\frac{1}{2}}\) is \(\sqrt{49}\), which equals 7. Now the expression looks like this: \(7^{-1}x^{1}y^{2}xy^{\frac{1}{2}}\).
03

Apply Product of Powers Rule

Apply the product of powers rule to \((x^{1}xy^{\frac{1}{2}})\) and \((y^{2}y^{\frac{1}{2}})\) respectively. This rule states that \(a^{m}a^{n} = a^{m+n}\). Now, the expression simplifies to \(7^{-1}x^{2}y^{\frac{5}{2}}\).
04

Use Negative Exponent Property

The negative exponent property states that \(a^{-n} = \frac{1}{a^{n}}\). Thus, \(7^{-1}x^{2}y^{\frac{5}{2}}\) becomes \(\frac{x^{2}y^{\frac{5}{2}}}{7}\).

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Most popular questions from this chapter

Read the Blitzer Bonus on page \(47 .\) The future is now: You have the opportunity to explore the cosmos in a starship traveling near the speed of light. The experience will enable you to understand the mysteries of the universe in deeply personal ways, transporting you to unimagined levels of knowing and being. The downside: You return from your two-year journey to a futuristic world in which friends and loved ones are long gone. Do you explore space or stay here on Earth? What are the reasons for your choice?

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(8 x^{-6} y^{3}\right)^{\frac{1}{3}}\left(x^{\frac{5}{6}} y^{-\frac{1}{3}}\right)^{6} $$

What does it mean to factor completely?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$-x-x=-x+(-x)=0$$

Factor Completely. $$x^{4}-5 x^{2} y^{2}+4 y^{4}$$
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