We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > College Algebra

Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

Example 9: Simplifying Exponential Expressions

Simplify each expression and write the answer with positive exponents only.
  1. (6m2n1)3{\left(6{m}^{2}{n}^{-1}\right)}^{3}
  2. 175174173{17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}
  3. (u1vv1)2{\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}
  4. (2a3b1)(5a2b2)\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)
  5. (x22)4(x22)4{\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}
  6. (3w2)5(6w2)2\frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}

Solution

  1. (6m2n1)3=(6)3(m2)3(n1)3The power of a product rule=63m23n13The power rule= 216m6n3Simplify.=216m6n3The negative exponent rule\begin{array}{cccc}\hfill {\left(6{m}^{2}{n}^{-1}\right)}^{3}& =& {\left(6\right)}^{3}{\left({m}^{2}\right)}^{3}{\left({n}^{-1}\right)}^{3}\hfill & \text{The power of a product rule}\hfill \\ & =& {6}^{3}{m}^{2\cdot 3}{n}^{-1\cdot 3}\hfill & \text{The power rule}\hfill \\ & =& \text{ }216{m}^{6}{n}^{-3}\hfill & \text{Simplify}.\hfill \\ & =& \frac{216{m}^{6}}{{n}^{3}}\hfill & \text{The negative exponent rule}\hfill \end{array}
  2. 175174173=17543The product rule=172Simplify.=1172 or 1289The negative exponent rule\begin{array}{cccc}\hfill {17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}& =& {17}^{5 - 4-3}\hfill & \text{The product rule}\hfill \\ & =& {17}^{-2}\hfill & \text{Simplify}.\hfill \\ & =& \frac{1}{{17}^{2}}\text{ or }\frac{1}{289}\hfill & \text{The negative exponent rule}\hfill \end{array}
  3. (u1vv1)2=(u1v)2(v1)2The power of a quotient rule=u2v2v2The power of a product rule=u2v2(2)The quotient rule=u2v4Simplify.=v4u2The negative exponent rule\begin{array}{cccc}\hfill {\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}& =& \frac{{\left({u}^{-1}v\right)}^{2}}{{\left({v}^{-1}\right)}^{2}}\hfill & \text{The power of a quotient rule}\hfill \\ & =& \frac{{u}^{-2}{v}^{2}}{{v}^{-2}}\hfill & \text{The power of a product rule}\hfill \\ & =& {u}^{-2}{v}^{2-\left(-2\right)}& \text{The quotient rule}\hfill \\ & =& {u}^{-2}{v}^{4}\hfill & \text{Simplify}.\hfill \\ & =& \frac{{v}^{4}}{{u}^{2}}\hfill & \text{The negative exponent rule}\hfill \end{array}
  4. (2a3b1)(5a2b2)=25a3a2b1b2Commutative and associative laws of multiplication=10a32b1+2The product rule=10abSimplify.\begin{array}{cccc}\hfill \left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)& =& -2\cdot 5\cdot {a}^{3}\cdot {a}^{-2}\cdot {b}^{-1}\cdot {b}^{2}\hfill & \text{Commutative and associative laws of multiplication}\hfill \\ & =& -10\cdot {a}^{3 - 2}\cdot {b}^{-1+2}\hfill & \text{The product rule}\hfill \\ & =& -10ab\hfill & \text{Simplify}.\hfill \end{array}
  5. (x22)4(x22)4=(x22)44The product rule= (x22)0Simplify.=1The zero exponent rule\begin{array}{cccc}\hfill {\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}& =& {\left({x}^{2}\sqrt{2}\right)}^{4 - 4}\hfill & \text{The product rule}\hfill \\ & =& \text{ }{\left({x}^{2}\sqrt{2}\right)}^{0}\hfill & \text{Simplify}.\hfill \\ & =& 1\hfill & \text{The zero exponent rule}\hfill \end{array}
  6. (3w2)5(6w2)2=(3)5(w2)5(6)2(w2)2The power of a product rule=35w2562w22The power rule=243w1036w4Simplify.=27w10(4)4The quotient rule and reduce fraction=27w144Simplify.\begin{array}{cccc}\hfill \frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}& =& \frac{{\left(3\right)}^{5}\cdot {\left({w}^{2}\right)}^{5}}{{\left(6\right)}^{2}\cdot {\left({w}^{-2}\right)}^{2}}\hfill & \text{The power of a product rule}\hfill \\ & =& \frac{{3}^{5}{w}^{2\cdot 5}}{{6}^{2}{w}^{-2\cdot 2}}\hfill & \text{The power rule}\hfill \\ & =& \frac{243{w}^{10}}{36{w}^{-4}}\hfill & \text{Simplify}.\hfill \\ & =& \frac{27{w}^{10-\left(-4\right)}}{4}\hfill & \text{The quotient rule and reduce fraction}\hfill \\ & =& \frac{27{w}^{14}}{4}\hfill & \text{Simplify}.\hfill \end{array}

Try It 9

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

a. (2uv2)3{\left(2u{v}^{-2}\right)}^{-3} b. x8x12x{x}^{8}\cdot {x}^{-12}\cdot x c. (e2f3f1)2{\left(\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2} d. (9r5s3)(3r6s4)\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right) e. (49tw2)3(49tw2)3{\left(\frac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\frac{4}{9}t{w}^{-2}\right)}^{3} f. (2h2k)4(7h1k2)2\frac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}

Solution

Licenses & Attributions