For all positive real numbers x and y, the expression \frac{(4x^3 y^2)^2}{\sqrt{16x^8 y^6}… — SAT

For all positive real numbers $x$ and $y$ , the expression $\frac{(4x^3 y^2)^2}{\sqrt{16x^8 y^6}}$ can be written in the equivalent form $ax^b y^c$ , where $a$ , $b$ , and $c$ are positive constants. What is the value of $a + b + c$ ?

Answer: 7

Answer

The expression simplifies to

4x^2 y^1

by applying the power of a product rule to the numerator to get

16x^6 y^4

, and simplifying the radical in the denominator to get

4x^4 y^3

. Dividing the terms yields

4x^2 y^1

. Identifying the coefficients and exponents gives

a = 4

b = 2

, and

c = 1

. The sum of these values is

4 + 2 + 1 = 7

Step-by-Step Solution

Simplify the numerator of the expression.

(4x^3 y^2)^2 = 16x^6 y^4

Apply the power of a product rule

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

and the power of a power rule

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

Simplify the denominator of the expression.

\sqrt{16x^8 y^6} = 4x^4 y^3

Take the square root of the coefficient and apply fractional exponent rules for positive variables:

\sqrt{16} = 4

\sqrt{x^8} = x^{8/2} = x^4

, and

\sqrt{y^6} = y^{6/2} = y^3

Divide the numerator by the denominator.

\frac{16x^6 y^4}{4x^4 y^3} = 4x^2 y^1

Divide the coefficients and subtract the exponents of the corresponding variables:

\frac{16}{4} = 4

x^{6-4} = x^2

, and

y^{4-3} = y^1

Identify the values of

a

b

, and

c

, and sum them.

a = 4

b = 2

c = 1

, and

a + b + c = 7

Match the simplified form

4x^2 y^1

ax^b y^c

to find

a

b

, and

c

, then calculate their sum.

Key Concept

Simplifying equivalent algebraic expressions using exponent rules and radical properties.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic