For all x > 0, the expression \left( \frac{x^{4/3} + 4x^{2/3} + 16}{x^{2/3} + 2x^{1/3} + 4… — SAT

For all $x > 0$ , the expression $\left( \frac{x^{4/3} + 4x^{2/3} + 16}{x^{2/3} + 2x^{1/3} + 4} + 2x^{1/3} \right)^3 - x^2 - 12x^{4/3} - 48x^{2/3}$ is equivalent to a constant $C$ . What is the value of $C$ ?

Answer: 64

Answer

The constant value is 64.

The expression inside the parentheses simplifies to

x^{2/3} + 4

after factoring the numerator as

(x^{2/3} + 2x^{1/3} + 4)(x^{2/3} - 2x^{1/3} + 4)

and canceling the common factor in the denominator. Cubing

x^{2/3} + 4

yields

x^2 + 12x^{4/3} + 48x^{2/3} + 64

. Subtracting the remaining terms

x^2 + 12x^{4/3} + 48x^{2/3}

from this expansion results in the constant value 64.

Step-by-Step Solution

Substitute

u = x^{1/3}

into the expression to simplify the fractional exponents.

The terms become

x^{1/3} = u

x^{2/3} = u^2

x^{4/3} = u^4

, and

x^2 = u^6

. The expression inside the parentheses is rewritten as

\frac{u^4 + 4u^2 + 16}{u^2 + 2u + 4} + 2u

Using a temporary variable

u

simplifies the algebraic factoring and manipulation of terms with fractional exponents.

Factor the numerator

u^4 + 4u^2 + 16

by completing the square.

u^4 + 4u^2 + 16 = (u^2 + 4)^2 - 4u^2 = (u^2 + 2u + 4)(u^2 - 2u + 4)

Expressing the quartic polynomial as a difference of squares allows it to be factored into two quadratic polynomials.

Simplify the rational expression and add

2u

\frac{(u^2 + 2u + 4)(u^2 - 2u + 4)}{u^2 + 2u + 4} + 2u = (u^2 - 2u + 4) + 2u = u^2 + 4

Canceling the common factor

u^2 + 2u + 4

in the numerator and denominator simplifies the expression inside the parentheses to

u^2 + 4

Substitute

u = x^{1/3}

back into

u^2 + 4

and cube the expression.

(x^{2/3} + 4)^3 = (x^{2/3})^3 + 3(x^{2/3})^2(4) + 3(x^{2/3})(16) + 64 = x^2 + 12x^{4/3} + 48x^{2/3} + 64

Applying the binomial expansion formula

(A + B)^3 = A^3 + 3A^2B + 3AB^2 + B^3

expands the cubed expression.

Subtract the remaining terms from the expanded expression.

(x^2 + 12x^{4/3} + 48x^{2/3} + 64) - x^2 - 12x^{4/3} - 48x^{2/3} = 64

Subtracting the variable terms cancels them out entirely, leaving the constant value 64.

Key Concept

Factoring quartic polynomials using the difference of squares and simplifying rational expressions with fractional exponents.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic