Combine positive and negative integer indices

Simplify x⁻² × x⁵. Assume x ≠ 0

For x⁻² divided by xⁿ, the resulting index is −2−n. Change the denominator index. The numberline shows the resulting index, not the value of x

Set n to −5. What is the resulting index?

Keep the signs when using index laws

Apply the same exponent laws to integer indices. Combine numerical coefficients separately. Every variable in these exercises is nonzero

1. 1Add indices for a product The same rule applies to positive, zero and negative integer indices \(\displaystyle x^{-2}x^5=x^{-2+5}=x^3\)
2. 2Subtract the denominator index Subtracting a negative index is different from subtracting its magnitude \(\displaystyle \frac{x^{-2}}{x^{-5}}=x^{-2-(-5)}=x^3\)
3. 3Multiply inside a power Multiply each inner index by the outer index before combining factors \(\displaystyle (x^{-2})^3=x^{-6}\)
4. 4Rewrite negative results A negative final index becomes a positive index in the denominator. All original variables remain nonzero \(\displaystyle 2x^{-3}y^2=\frac{2y^2}{x^3}\)

Quotient, x ≠ 0

\(\displaystyle \frac{x^{-2}}{x^3}\)

What is the signed index of x after simplifying this quotient?

Quotient

\(\displaystyle \frac{x^{-2}}{x^{-5}}\)

Which calculation correctly combines the indices? Assume x ≠ 0.

A simplified answer no longer shows a denominator. Can the original nonzero restriction be dropped?