When the algebraic expression \frac{(3x^{-2} y^4)^3}{9x^{-5} y^2} is fully simplified to t… — Armed Services Vocational Aptitude Battery (ASVAB)

When the algebraic expression

\frac{(3x^{-2} y^4)^3}{9x^{-5} y^2}

is fully simplified to the form

C x^a y^b

(where

C

a

, and

b

are constants), what is the value of

b

Cevap: 10 / ten

Cevap

By distributing the outer exponent of 3 in the numerator using multiplication (

4 \cdot 3

) to get

y^{12}

, and then subtracting the denominator's exponent of 2 using the quotient rule (

12 - 2

), the final exponent on the variable

y

is determined to be 10.

Adım Adım Çözüm

Apply the power of a product and power of a power rules to the numerator.

(3x^{-2}y^4)^3 = 3^3 \cdot (x^{-2})^3 \cdot (y^4)^3 = 27x^{-6}y^{12}

When raising a product to a power, the outer power applies to each individual factor inside the parentheses. Exponents are multiplied when raising a power to another power.

Substitute the expanded numerator back into the fractional expression.

\frac{27x^{-6}y^{12}}{9x^{-5}y^2}

This sets up the expression for applying the quotient rule between the numerator and denominator.

Apply the quotient rule by dividing the coefficients and subtracting the exponents of like bases.

Coefficient:

27 \div 9 = 3

Base

x

x^{-6 - (-5)} = x^{-6 + 5} = x^{-1}

Base

y

y^{12 - 2} = y^{10}

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

Combine the simplified terms and identify the exponent

b

The simplified expression is

3x^{-1}y^{10}

. Comparing this to the target form

Cx^a y^b

, the value of

b

10

This isolates the specific value requested by the question stem.

Anahtar Kavram

Power of a Power and Quotient of Powers Rules

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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