Question

rewrite the expression using only positive exponents, and simplify. (assume that any variables in the expression are nonzero.)
(ab)^{-9}(a^{9}b^{9})^{-1}

rewrite the expression using only positive exponents, and simplify. (assume that any variables in the expression are nonzero.)
u^{9}(2u^{-8}v^{0})(7v)^{0}

Explanation:

First Expression: \((ab)^{-9}(a^{9}b^{9})^{-1}\)

Step 1: Apply the power of a product rule \((xy)^n = x^n y^n\) and negative exponent rule \(x^{-n}=\frac{1}{x^n}\)

\((ab)^{-9}=a^{-9}b^{-9}\) and \((a^{9}b^{9})^{-1}=a^{-9}b^{-9}\)
So the expression becomes \(a^{-9}b^{-9}\cdot a^{-9}b^{-9}\)

Step 2: Apply the product rule for exponents \(x^m \cdot x^n = x^{m + n}\)

For \(a\): \(-9+(-9)=-18\), for \(b\): \(-9+(-9)=-18\)
So we have \(a^{-18}b^{-18}\)

Step 3: Convert negative exponents to positive using \(x^{-n}=\frac{1}{x^n}\)

\(a^{-18}b^{-18}=\frac{1}{a^{18}b^{18}}\)

Second Expression: \(u^{9}(2u^{-8}v^{0})(7v)^{0}\)

Step 1: Recall that any non - zero number to the power of 0 is 1, so \(v^{0}=1\) and \((7v)^{0}=1\)

The expression simplifies to \(u^{9}(2u^{-8}\cdot1)\cdot1\)

Step 2: Apply the product rule for exponents \(x^m \cdot x^n = x^{m + n}\) and multiply the coefficients

First, multiply the coefficients: \(2\times1\times1 = 2\)
For the \(u\) terms: \(u^{9}\cdot u^{-8}=u^{9+(-8)}=u^{1}=u\)
So the expression is \(2u\)

Answer:

For \((ab)^{-9}(a^{9}b^{9})^{-1}\): \(\frac{1}{a^{18}b^{18}}\)

For \(u^{9}(2u^{-8}v^{0})(7v)^{0}\): \(2u\)