Question

which expression is equivalent to (4x^{-2}+3x^{-5})?
a. (7x^{-7})
b. (12x^{10})
c. (\frac{7}{x^{7}})
d. (\frac{4x^{3}+3}{x^{5}})
e. (\frac{4x^{5}+3}{x^{7}})

Explanation:

Step1: Rewrite negative exponents

Recall $x^{-n}=\frac{1}{x^n}$, so:
$4x^{-2} + 3x^{-5} = \frac{4}{x^2} + \frac{3}{x^5}$

Step2: Find common denominator

The common denominator is $x^5$. Adjust the first term:
$\frac{4}{x^2} = \frac{4x^{5-2}}{x^5} = \frac{4x^3}{x^5}$

Step3: Combine the fractions

Add the two fractions over the common denominator:
$\frac{4x^3}{x^5} + \frac{3}{x^5} = \frac{4x^3 + 3}{x^5}$

Answer:

D. $\frac{4x^3+3}{x^5}$