6.7 Integer exponents and scientific notation  (Page 3/10)

Solution

1. ⓐ
   $\begin{array}{cccc}& & & x^{\mathrm{-4}}·x^6\hfill \\ \text{Use the Product Property,}{a}^m·a^n=a^{m+n}.\hfill & & & x^{\mathrm{-4}+6}\hfill \\ \text{Simplify.}\hfill & & & x^2\hfill \end{array}$
   
   
2. ⓑ
   $\begin{array}{cccc}& & & y^{\mathrm{-6}}·y^4\hfill \\ \text{Notice the same bases, so add the exponents.}\hfill & & & y^{\mathrm{-6}+4}\hfill \\ \text{Simplify.}\hfill & & & y^{\mathrm{-2}}\hfill \\ \text{Use the definition of a negative exponent,}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & \frac{1}{y^2}\hfill \end{array}$
   
   
3. ⓒ
   $\begin{array}{cccc}& & & z^{\mathrm{-5}}·z^{\mathrm{-3}}\hfill \\ \text{Add the exponents, since the bases are the same.}\hfill & & & z^{\mathrm{-5}-3}\hfill \\ \text{Simplify.}\hfill & & & z^{\mathrm{-8}}\hfill \\ \begin{array}{c}\text{Take the reciprocal and change the sign of the exponent,}\hfill \\ \text{using the definition of a negative exponent.}\hfill \end{array}\hfill & & & \frac{1}{z^8}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \left(m^4{n}^{\mathrm{-3}}\right)\left(m^{\mathrm{-5}}{n}^{\mathrm{-2}}\right)\hfill \\ \text{Use the Commutative Property to get like bases together.}\hfill & & & m^4{m}^{\mathrm{-5}}·n^{\mathrm{-2}}{n}^{\mathrm{-3}}\hfill \\ \text{Add the exponents for each base.}\hfill & & & m^{\mathrm{-1}}·n^{\mathrm{-5}}\hfill \\ \text{Take reciprocals and change the signs of the exponents.}\hfill & & & \frac{1}{m^1}·\frac{1}{n^5}\hfill \\ \text{Simplify.}\hfill & & & \frac{1}{m{n}^5}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \left(2x^{\mathrm{-6}}{y}^8\right)\left(\mathrm{-5}{x}^5{y}^{\mathrm{-3}}\right)\hfill \\ \text{Rewrite with the like bases together.}\hfill & & & 2\left(\mathrm{-5}\right)·\left(x^{\mathrm{-6}}{x}^5\right)·\left(y^8{y}^{\mathrm{-3}}\right)\hfill \\ \text{Multiply the coefficients and add the exponents of each variable.}\hfill & & & \mathrm{-10}·x^{\mathrm{-1}}·y^5\hfill \\ \text{Use the definition of a negative exponent,}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & \mathrm{-10}·\frac{1}{x^1}·y^5\hfill \\ \text{Simplify.}\hfill & & & \frac{\mathrm{-10}{y}^5}{x}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & {\left(6k^3\right)}^{\mathrm{-2}}\hfill \\ \text{Use the Product to a Power Property,}{\left(ab\right)}^m=a^m{b}^m.\hfill & & & {\left(6\right)}^{\mathrm{-2}}{\left(k^3\right)}^{\mathrm{-2}}\hfill \\ \text{Use the Power Property,}{\left(a^m\right)}^n=a^{m·n}.\hfill & & & 6^{\mathrm{-2}}{k}^{\mathrm{-6}}\hfill \\ \text{Use the Definition of a Negative Exponent,}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & \frac{1}{6^2}·\frac{1}{k^6}\hfill \\ \text{Simplify.}\hfill & & & \frac{1}{36k^6}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & {\left(5x^{\mathrm{-3}}\right)}^2\hfill \\ \\ \\ \text{Use the Product to a Power Property,}{\left(ab\right)}^m=a^m{b}^m.\hfill & & & 5^2{\left(x^{\mathrm{-3}}\right)}^2\hfill \\ \\ \\ \begin{array}{c}\text{Simplify}{5}^2\text{and multiply the exponents of}x\text{using the Power}\hfill \\ \text{Property,}{\left(a^m\right)}^n=a^{m·n}.\hfill \end{array}\hfill & & & 25·x^{\mathrm{-6}}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite}{x}^{\mathrm{-6}}\text{by using the Definition of a Negative Exponent,}\hfill \\ a^{\text{−}n}=\frac{1}{a^n}.\hfill \end{array}\hfill & & & 25·\frac{1}{x^6}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \frac{25}{x^6}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \frac{r^5}{r^{\mathrm{-4}}}\hfill \\ \text{Use the Quotient Property,}\frac{a^m}{a^n}=a^{m-n}.\hfill & & & r^{5-\left(\mathrm{-4}\right)}\hfill \\ \text{Simplify.}\hfill & & & r^9\hfill \end{array}$