which expression is equivalent to $4x^{-2} + 3x^{-5}$?\na. $7x^{-7}$\nb. $12x^{10}$\nc. $\frac{7}{x^7}$\nd…

Explanation:

Step1: Recall Negative Exponent Rule

The negative exponent rule states that ( a^{-n} = \frac{1}{a^n} ) (where ( a \neq 0 ) 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 gives ( 4\times\frac{1}{x^{2}}=\frac{4}{x^{2}} ).

For ( 3x^{-5} ), applying the rule gives ( 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} ) (since ( x^{5} ) is a multiple of ( x^{2} )).

We rewrite ( \frac{4}{x^{2}} ) with the denominator ( x^{5} ) by multiplying the numerator and denominator by ( x^{3} ) (because ( x^{2}\times x^{3}=x^{5} )):

( \frac{4}{x^{2}}=\frac{4\times x^{3}}{x^{2}\times x^{3}}=\frac{4x^{3}}{x^{5}} )

Step3: Add the Fractions

Now that both fractions have the same denominator, we can add them:

( \frac{4x^{3}}{x^{5}}+\frac{3}{x^{5}}=\frac{4x^{3} + 3}{x^{5}} )

Answer:

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