The algebraic expression \frac{(2x^4 y^3)^3}{4x^5 y^{-2}} can be simplified to the form kx… — Armed Services Vocational Aptitude Battery (ASVAB)

The algebraic expression

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

can be simplified to the form

kx^a y^b

, where

k

is the numerical coefficient, and

a

and

b

are the exponents. What is the value of

a + b

Cevap: 18 / 18.0

By applying the power rule, the numerator becomes

8x^{12}y^9

. Dividing this by the denominator

4x^5y^{-2}

yields a numerical coefficient of

8/4=2

. For the

x

terms, applying the quotient rule gives

12 - 5 = 7

. For the

y

terms,

9 - (-2) = 11

. The simplified expression is

2x^7y^{11}

. The exponents are

a=7

and

b=11

, so their sum is

7 + 11 = 18

Apply the power of a product rule to the numerator.

(2^3)(x^{4 \times 3})(y^{3 \times 3}) = 8x^{12}y^9

To distribute the outer exponent 3 to all factors inside the parentheses.

Divide the numerical coefficients.

\frac{8}{4} = 2

To simplify the constants to find

k

Apply the quotient rule for the variable

x

x^{12 - 5} = x^7

When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator. This gives

a = 7

Apply the quotient rule for the variable

y

, accounting for the negative exponent.

y^{9 - (-2)} = y^{9 + 2} = y^{11}

Subtracting a negative exponent is equivalent to adding. This gives

b = 11

Calculate the sum of the exponents

a

and

b

7 + 11 = 18

To answer the specific question asked in the prompt.

Applying the power of a product rule and the quotient rule for exponents, paying special attention to subtracting negative exponents.

Soru