If x and y are positive real numbers, what is the simplified form of the expression below?… — ACT

If $x$ and $y$ are positive real numbers, what is the simplified form of the expression below?

\frac{4 x^{1/2} y^3 \cdot x^{1/3} y^{-1}}{2 x^{-1} y}

The simplified form of the expression is

2x^{11/6}y

The correct answer is

2x^{11/6}y

. First, simplify the constant coefficients to obtain

\frac{4}{2} = 2

. In the numerator, combine the bases by adding exponents:

x^{1/2} \cdot x^{1/3} = x^{1/2 + 1/3} = x^{5/6}

, and

y^3 \cdot y^{-1} = y^{3 - 1} = y^2

. Next, divide by the terms in the denominator using the quotient rule:

\frac{x^{5/6}}{x^{-1}} = x^{5/6 - (-1)} = x^{11/6}

, and

\frac{y^2}{y^1} = y^{2 - 1} = y

. This yields the fully simplified expression

2x^{11/6}y

Simplify the coefficients of the fraction.

\frac{4}{2} = 2

Dividing the numerical constants in the numerator and denominator simplifies the constant multiplier of the expression.

Combine the

x

terms in the numerator using the product rule.

x^{1/2} \cdot x^{1/3} = x^{1/2 + 1/3} = x^{5/6}

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

\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Combine the

y

terms in the numerator using the product rule.

y^3 \cdot y^{-1} = y^{3 + (-1)} = y^2

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

3 - 1 = 2

Simplify the

x

terms in the fraction using the quotient rule.

\frac{x^{5/6}}{x^{-1}} = x^{5/6 - (-1)} = x^{5/6 + 1} = x^{11/6}

When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.

Simplify the

y

terms in the fraction using the quotient rule.

\frac{y^2}{y^1} = y^{2 - 1} = y

When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.

Combine the simplified components.

2x^{11/6}y

Multiply the simplified coefficient,

x

term, and

y

term together to get the final simplified expression.

Properties of Exponents in Algebraic Expressions

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