If x and y are positive numbers, which of the following is equivalent to the expression \f… — SAT

If $x$ and $y$ are positive numbers, which of the following is equivalent to the expression $\frac{\sqrt{36x^5 y^3}}{3x(xy^3)^{1/2}}$ ?

The expression is equivalent to

2x

To find the equivalent expression, simplify the numerator and the denominator separately. The numerator

\sqrt{36x^5 y^3}

simplifies to

6x^{5/2}y^{3/2}

. The denominator

3x(xy^3)^{1/2}

simplifies to

3x^{3/2}y^{3/2}

. Dividing the simplified numerator by the simplified denominator yields

\frac{6x^{5/2}y^{3/2}}{3x^{3/2}y^{3/2}} = 2x

Convert the radical expression in the numerator to an expression with fractional exponents.

\sqrt{36x^5 y^3} = (36x^5 y^3)^{1/2} = 6x^{5/2}y^{3/2}

The square root of a product is the product of the square roots of each factor, where

\sqrt{a} = a^{1/2}

Apply exponent rules to simplify the denominator.

3x(xy^3)^{1/2} = 3x^1 \cdot x^{1/2} y^{3/2} = 3x^{3/2}y^{3/2}

When multiplying terms with the same base, add their exponents (

x^a \cdot x^b = x^{a+b}

Divide the simplified numerator by the simplified denominator.

\frac{6x^{5/2}y^{3/2}}{3x^{3/2}y^{3/2}} = 2x^{5/2 - 3/2}y^{3/2 - 3/2} = 2x^1 y^0 = 2x

When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator (

\frac{x^a}{x^b} = x^{a-b}

Equivalent Algebraic Expressions

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