Problem 111 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(49^{-\frac{1}{2}} (x^{-2})^{-\frac{1}{2}} (y^{4})^{-\frac{1}{2}} (x y^{\frac{1}{2}})\).

Step 2: 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: \(7^{-1}x^{1}y^{2}xy^{\frac{1}{2}}\).

Step 3: 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 \(7^{-1}x^{2}y^{\frac{5}{2}}\).

Step 4: 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 \(\frac{x^{2}y^{\frac{5}{2}}}{7}\).