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Write $4y^{-6}$ without using negative exponents.

Write $\dfrac{ 3 x^{-4} }{ yz^{-2} }$ without using negative exponents.

Simplify $\left(-5x^{-5}\right)\left(-8x^4\right)$ and write it without using negative exponents.

Solution

Always remember that an exponent only applies to what it is touching. In the expression $4y^{-6}\text{,}$ only the $y$ has an exponent of $-6\text{.}$
$\begin{aligned} 4y^{-6} \amp= 4\cdot \frac{1}{y^{6}}\\ \amp= \frac{4}{y^6} \end{aligned}$

Negative exponents tell us to move some variables between the numerator and denominator to make the exponents positive.
$\frac{ 3 \highlight{x^{-4}} }{ y\highlight{z^{-2}} }=\frac{ 3 \highlight{z^{2}} }{ y\highlight{x^{4}} }$
Notice that the factors of $3$ and $y$ did not move, as both of those factors had positive exponents.

The product of powers rule still applies, and we can add exponents even when one or both are negative:
$\begin{aligned} \left(-5x^{-5}\right)\left(-8x^4\right)\amp=(-5)(-8)x^{-5}x^4\\ \amp=40x^{-5+4}\\ \amp=40x^{-1}\\ \amp=\frac{40}{x^1}\\ \amp=\frac{40}{x} \end{aligned}$