skaterkevin8n8

2022-03-15

When you are asked to factor with negative exponents, how do you know what your gcf is? Consider ${x}^{-3}+2{x}^{-4}$. What is the gcf? How does this relate to when you are asked to factor ${x}^{3}+2{x}^{4}$?

That’s a great question. When you factor with positive exponents, you look for the largest factor that is found in all the addends. That “largest” can never be higher that the least exponent. So, if you consider ${x}^{3}+2{x}^{4}$, that factor is ${x}^{3}$, and the result ${x}^{3}\left(1+2x\right)$.
When you are dealing with negative exponents, the answer is the same. Just remember that -3 is greater than -4. So, your gcf in this case will be ${x}^{-4}$. The factored result, then, is ${x}^{-4}\left(x+2\right)$.

conduchafr4

You can factor negative exponents similar to postive ones:
${x}^{-3}+2{x}^{-4}={x}^{-3}\left(1+2{x}^{-1}\right)$
And ${x}^{3}+2{x}^{4}={x}^{3}\left(1+2x\right)$
${a}^{b}\cdot {a}^{c}={a}^{b+c}$
${a}^{-b}=\frac{1}{{a}^{b}}$
Thus the first line can also read
$\frac{1}{{x}^{3}}+\frac{2}{{x}^{4}}=\frac{1}{{x}^{3}}\cdot \left(1+\frac{2}{x}\right)$

Do you have a similar question?