Question

which expression is equivalent to $4x^{-2}+3x^{-5}$?
\\(\bigcirc\\) a $7x^{-7}$
\\(\bigcirc\\) b $12x^{10}$
\\(\bigcirc\\) c $\frac{7}{x^7}$
\\(\bigcirc\\) d $\frac{4x^3 + 3}{x^5}$
\\(\bigcirc\\) e $\frac{4x^3 + 1}{x^2}$

Explanation:

Step1: Recall negative exponent rule

The negative exponent rule states that \( a^{-n}=\frac{1}{a^{n}} \) (where \( a
eq0 \) and \( n \) is a positive integer). So we can rewrite each term in the expression \( 4x^{-2}+3x^{-5} \) using this rule.
For \( 4x^{-2} \), applying the rule, we get \( 4\times\frac{1}{x^{2}}=\frac{4}{x^{2}} \).
For \( 3x^{-5} \), applying the rule, we get \( 3\times\frac{1}{x^{5}}=\frac{3}{x^{5}} \).

Step2: Find a common denominator

To add the two fractions \( \frac{4}{x^{2}} \) and \( \frac{3}{x^{5}} \), we need a common denominator. The least common denominator of \( x^{2} \) and \( x^{5} \) is \( x^{5} \).
We rewrite \( \frac{4}{x^{2}} \) with denominator \( x^{5} \) by multiplying the numerator and denominator by \( x^{3} \):
\( \frac{4}{x^{2}}=\frac{4\times x^{3}}{x^{2}\times x^{3}}=\frac{4x^{3}}{x^{5}} \)

Step3: Add the fractions

Now we can add the two fractions:
\( \frac{4x^{3}}{x^{5}}+\frac{3}{x^{5}}=\frac{4x^{3} + 3}{x^{5}} \)

Answer:

\(\frac{4x^{3}+3}{x^{5}}\) (corresponding to option D)