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Section 14.1 Negative Powers Prep Activities
Prep Activity 14.1.1 .
Let’s discover another exponent rule.
(a) Use repeated multiplication to rewrite each of the following expressions, then simplify by canceling. Remember, if you cancel everything from the numerator of the fraction, then you’re left a
\(1\) in the numerator.
(i) (ii) (b) Use the Quotient Rule for Exponents to rewrite each of the following expressions as
\(x^{\text{power}}\text{.}\)
(i) (ii) (c)
Explain why the last two parts support the reasonableness of the rule
\begin{equation*}
\frac{1}{x^n} = x^{-n}\text{.}
\end{equation*}
Prep Activity 14.1.2 .
Let’s practice using our new rule, along with other exponent rules, to rewrite expressions.
(a) Rewrite as
\(ax^\text{power}\text{:}\) \(\frac{6}{x}\)
(b) Rewrite as
\(ax^\text{power}\text{:}\) \(\frac{4x^3}{8x^6}\)
(c) Rewrite without any negative exponents:
\(6^{-1}x^3x^{-5}\)
(d) Rewrite without any negative exponents:
\(\left(4x^{-6}\right)^{-2}\)
Prep Activity 14.1.3 .
In many cases, the best way to deal with negative exponents and/or a variable in the denominator in an equation is to move the variable to the other side of the equation using multiplication. For example:
\begin{align*}
9 \amp= \frac{5}{x^6}\\
x^6 \cdot 9 \amp= \frac{5}{x^6} \cdot x^6 \\
9x^6 \amp= 5 \\
x^6 \amp= \frac{5}{9} \\
x \amp = \pm \sqrt[6]{\frac{5}{9}}
\end{align*}
Use this strategy to solve each of the equations below.
(a) (b)
Prep Activity 14.1.4 .
You’ll need to be able to do the following things for this lesson. Rate how confident you are on a scale of 1 - 5 (1 = not confident and 5 = very confident).
Table 14.1.1.
Use the equality \(\frac{1}{x^n} = x^{-n}\) to rewrite expressions.
Solve equations involving negative exponents.
