Simplifying Exponential Expressions

Simplify each expression and write the answer with positive exponents only.

ⓐ ${(6 m^{2} n^{−1})}^{3}$
ⓑ $17^{5} \cdot 17^{−4} \cdot 17^{−3}$
ⓒ ${(\frac{u^{−1} v}{v^{−1}})}^{2}$
ⓓ $(−2 a^{3} b^{−1}) (5 a^{−2} b^{2})$
ⓔ ${(x^{2} \sqrt{2})}^{4} {(x^{2} \sqrt{2})}^{−4}$
ⓕ $\frac{{(3 w^{2})}^{5}}{{(6 w^{−2})}^{2}}$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ
  $\begin{matrix} {(6 m^{2} n^{−1})}^{3} & = & {(6)}^{3} {(m^{2})}^{3} {(n^{−1})}^{3} & The power of a product rule \\ = & 6^{3} m^{2 \cdot 3} n^{−1 \cdot 3} & The power rule \\ = & 216m^{6}n^{−3} & Simplify . \\ = & \frac{216 m^{6}}{n^{3}} & The negative exponent rule \end{matrix}$
2. ⓑ
  $\begin{matrix} 17^{5} \cdot 17^{−4} \cdot 17^{−3} & = & 17^{5 - 4 - 3} & The product rule \\ = & 17^{−2} & Simplify . \\ = & \frac{1}{17^{2}} or \frac{1}{289} & The negative exponent rule \end{matrix}$
3. ⓒ
  $\begin{matrix} {(\frac{u^{−1} v}{v^{−1}})}^{2} & = & \frac{{(u^{−1} v)}^{2}}{{(v^{−1})}^{2}} & The power of a quotient rule \\ = & \frac{u^{−2} v^{2}}{v^{−2}} & The power of a product rule \\ = & u^{−2} v^{2 - (−2)} & The quotient rule \\ = & u^{−2} v^{4} & Simplify . \\ = & \frac{v^{4}}{u^{2}} & The negative exponent rule \end{matrix}$
4. ⓓ
  $\begin{matrix} (−2 a^{3} b^{- 1}) (5 a^{−2} b^{2}) & = & −2 \cdot 5 \cdot a^{3} \cdot a^{−2} \cdot b^{−1} \cdot b^{2} & Commutative and associative laws of multiplication \\ = & −10 \cdot a^{3 - 2} \cdot b^{−1 + 2} & The product rule \\ = & −10 a b & Simplify . \end{matrix}$
5. ⓔ
  $\begin{matrix} {(x^{2} \sqrt{2})}^{4} {(x^{2} \sqrt{2})}^{−4} & = & {(x^{2} \sqrt{2})}^{4 - 4} & The product rule \\ = & {(x^{2} \sqrt{2})}^{0} & Simplify . \\ = & 1 & The zero exponent rule \end{matrix}$
6. ⓕ
  $\begin{matrix} \frac{{(3 w^{2})}^{5}}{{(6 w^{−2})}^{2}} & = & \frac{{(3)}^{5} \cdot {(w^{2})}^{5}}{{(6)}^{2} \cdot {(w^{−2})}^{2}} & The power of a product rule \\ = & \frac{3^{5} w^{2 \cdot 5}}{6^{2} w^{−2 \cdot 2}} & The power rule \\ = & \frac{243 w^{10}}{36 w^{−4}} & Simplify . \\ = & \frac{27 w^{10 - (−4)}}{4} & The quotient rule and reduce fraction \\ = & \frac{27 w^{14}}{4} & Simplify . \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Simplify each expression and write the answer with positive exponents only.

ⓐ ${(2 u v^{- 2})}^{−3}$
ⓑ $x^{8} \cdot x^{−12} \cdot x$
ⓓ $(9 r^{−5} s^{3}) (3 r^{6} s^{−4})$
ⓔ ${(\frac{4}{9} t w^{−2})}^{−3} {(\frac{4}{9} t w^{−2})}^{3}$
ⓕ $\frac{{(2 h^{2} k)}^{4}}{{(7 h^{−1} k^{2})}^{2}}$

How did it go?

Solution

Your turn