For all positive real numbers x and y, the expression \frac{(x^{3} y^{2})^{2/3} \cdot (x^{… — SAT

For all positive real numbers $x$ and $y$ , the expression $\frac{(x^{3} y^{2})^{2/3} \cdot (x^{-1} y^{4})^{1/6}}{(x^2 y)^{1/2}}$ can be written in the equivalent form $x^a y^b$ , where $a$ and $b$ are constants. What is the value of $6a + 2b$ ?

Answer: 8

Answer

The correct answer is 8.

Applying the rules of exponents systematically yields the simplified expression

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

. By setting

a = \frac{5}{6}

and

b = \frac{3}{2}

, the linear combination

6a + 2b

evaluates to

6\left(\frac{5}{6}\right) + 2\left(\frac{3}{2}\right) = 5 + 3 = 8

Step-by-Step Solution

Apply the power of a power rule to the first term in the numerator.

(x^3 y^2)^{2/3} = x^{3 \cdot \frac{2}{3}} y^{2 \cdot \frac{2}{3}} = x^2 y^{4/3}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

Apply the power of a power rule to the second term in the numerator.

(x^{-1} y^4)^{1/6} = x^{-1 \cdot \frac{1}{6}} y^{4 \cdot \frac{1}{6}} = x^{-1/6} y^{2/3}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

Multiply the two simplified terms in the numerator.

(x^2 y^{4/3})(x^{-1/6} y^{2/3}) = x^{2 - \frac{1}{6}} y^{\frac{4}{3} + \frac{2}{3}} = x^{11/6} y^2

When multiplying expressions with the same base, add their exponents.

Simplify the denominator.

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

To raise a product to a power, raise each factor to that power by multiplying their exponents.

Divide the numerator by the denominator.

\frac{x^{11/6} y^2}{x y^{1/2}} = x^{\frac{11}{6} - 1} y^{2 - \frac{1}{2}} = x^{5/6} y^{3/2}

When dividing expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Identify the values of

a

and

b

and calculate

6a + 2b

a = \frac{5}{6}

b = \frac{3}{2}

, so

6a + 2b = 6\left(\frac{5}{6}\right) + 2\left(\frac{3}{2}\right) = 5 + 3 = 8

Matching the simplified expression

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

x^a y^b

yields the values of the constants

a

and

b

, which are then used to calculate the required expression.

Key Concept

Simplifying rational expressions with fractional exponents using exponent rules.

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