For all real numbers x > 1, the expression below is equivalent to which of the following? … — ACT

For all real numbers $x > 1$ , the expression below is equivalent to which of the following?

\frac{\sqrt{x^8} - x^2 \cdot x^3}{x^2(1-x) - (x^2 - x)^2}

The expression is equivalent to

x

The correct answer is

x

. Simplifying the numerator yields

x^4 - x^5 = x^4(1-x)

. Simplifying the denominator by expanding both parts yields

(x^2 - x^3) - (x^4 - 2x^3 + x^2) = x^3 - x^4 = x^3(1-x)

. Dividing the numerator by the denominator and canceling the common non-zero term

(1-x)

leaves

\frac{x^4}{x^3}

, which simplifies to

x

Simplify the radical term in the numerator.

\sqrt{x^8} = x^4

Since

x > 1

, we can take the square root of

x^8

directly by dividing the exponent by 2:

(x^8)^{1/2} = x^{8/2} = x^4

Simplify the multiplication of exponential terms in the numerator.

x^2 \cdot x^3 = x^5

By exponent rules, when multiplying terms with the same base, we add the exponents:

x^{2+3} = x^5

Combine the terms in the numerator and factor out the greatest common factor.

Numerator =

x^4 - x^5 = x^4(1-x)

Factoring out

x^4

prepares the numerator for potential cancellation with the denominator.

Expand and simplify the terms in the denominator.

Denominator =

x^3(1-x)

Expanding the first term gives

x^2(1-x) = x^2 - x^3

. Expanding the squared binomial gives

(x^2-x)^2 = x^4 - 2x^3 + x^2

. Subtracting them yields

(x^2 - x^3) - (x^4 - 2x^3 + x^2) = x^3 - x^4 = x^3(1-x)

Divide the simplified numerator by the simplified denominator.

\frac{x^4(1-x)}{x^3(1-x)} = x

Since

x > 1

, the factor

(1-x)

is non-zero, allowing us to cancel it. Finally, dividing the remaining terms gives

\frac{x^4}{x^3} = x^{4-3} = x^1 = x

Simplifying rational expressions by factoring and applying exponent rules, including radical simplification and binomial expansion.

Question