For all positive values of x and y, the expression \frac{(3x^2 y^4)^3}{9x^4 y^7} can be wr… — SAT

For all positive values of $x$ and $y$ , the expression $\frac{(3x^2 y^4)^3}{9x^4 y^7}$ can be written in the form $a x^b y^c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $a + b + c$ ?

Answer: 10

Answer

The value of

a + b + c

10

To find the value of

a + b + c

, we simplify the given expression using the rules of exponents. First, apply the power of a product rule and power of a power rule to the numerator:

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

. Next, divide this by the denominator:

\frac{27x^6y^{12}}{9x^4y^7}

. Divide the coefficients to get

\frac{27}{9} = 3

, and apply the quotient rule for exponents to the variable terms:

x^{6-4} = x^2

and

y^{12-7} = y^5

. The fully simplified expression is

3x^2y^5

, which matches the form

a x^b y^c

. Comparing coefficients and exponents, we find

a = 3

b = 2

, and

c = 5

. Summing these values gives

3 + 2 + 5 = 10

Step-by-Step Solution

Simplify the numerator of the expression using exponent rules.

27x^6y^{12}

Applying the power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

(a^m)^n = a^{mn}

(3x^2 y^4)^3

gives

3^3 \cdot x^{2 \times 3} \cdot y^{4 \times 3} = 27x^6y^{12}

Divide the simplified numerator by the denominator.

3x^2y^5

Divide the coefficients (

27 \div 9 = 3

) and subtract the exponents of the corresponding variables using the quotient of powers rule

\frac{a^m}{a^n} = a^{m-n}

(

x^{6-4} = x^2

and

y^{12-7} = y^5

Identify the values of the constants

a

b

, and

c

from the simplified expression

3x^2y^5

a = 3

b = 2

, and

c = 5

Comparing the simplified expression

3x^2y^5

with the template

a x^b y^c

yields

a = 3

b = 2

, and

c = 5

Calculate the sum of

a

b

, and

c

10

Adding the identified constant values together:

3 + 2 + 5 = 10

Key Concept

Simplifying exponential expressions using exponent rules

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