Let’s explore exponent patterns with bases other than 10.
Is each statement true or false? Be prepared to explain your reasoning.
3^5 < 4^6
\left(\text- 3\right)^2 < 3^2
\left(\text- 3\right)^3 = 3^3
\left( \text- 5 \right) ^2 > \text- 5^2
Complete the table to show what it means to have an exponent of zero or a negative exponent.
Find an expression equivalent to \left(\frac{2}{3}\right)^{\text-3} but with positive exponents.
Find an expression equivalent to \left(\frac{4}{5}\right)^{\text-8} but with positive exponents.
What patterns do you notice when you start with a fraction to a negative power and rewrite it so that it has only positive powers? Show or explain your reasoning.
Lin, Noah, Diego, and Elena decide to test each other’s knowledge of exponents with bases other than 10. They each chose an expression to start with and then came up with a new list of expressions; some of which are equivalent to the original and some of which are not.
Choose 2 lists to analyze. For each list of expressions you choose to analyze, decide which expressions are not equivalent to the original. Be prepared to explain your reasoning.
(5^3)^{\text-3}
\text-5^9
\frac{5^{\text-6}}{5^3}
(5^3)^{\text-2}
\frac{5^{\text-4}}{5^{\text-5}}
5^{\text-4} \boldcdot 5^{\text-5}
3^5 \boldcdot 3^2
(3^5)^2
(3 \boldcdot 3)( 3 \boldcdot 3)( 3 \boldcdot 3)( 3 \boldcdot 3)( 3 \boldcdot 3)
\left(\frac{1}{3}\right)^{\text-10}
3^7 \boldcdot 3^3
\frac{3^{20}}{3^{10}}
\frac{3^{20}}{3^2}
\frac{x^8}{x^4}
x \boldcdot x \boldcdot x \boldcdot x
\frac{x^{\text-4}}{x^{\text-8}}
\frac{x^{\text-4}}{x^8}
(x^2)^2
4 \boldcdot x
x \boldcdot x^3
1
0
8^3 \boldcdot 8^{\text-3}
\frac{8^2}{8^2}
10^0
11^0
Earlier we focused on powers of 10 because 10 plays a special role in the decimal number system. But the exponent rules that we developed for 10 also work for other bases. For example, if 2^0=1 and 2^{\text -n} = \frac{1}{2^n}, then
\begin{align}2^m \boldcdot 2^n &= 2^{m+n} \\ \left(2^m\right)^n &= 2^{m \boldcdot n} \\ \frac{2^m}{2^n} &= 2^{m\text -n}. \end{align}
These rules also work for powers of numbers less than 1. For example, \left(\frac{1}{3}\right)^2 = \frac{1}{3} \boldcdot \frac{1}{3} and \left(\frac{1}{3}\right)^4 = \frac{1}{3} \boldcdot \frac{1}{3} \boldcdot \frac{1}{3} \boldcdot \frac{1}{3}. We can also check that \left(\frac{1}{3}\right)^2 \boldcdot \left(\frac{1}{3}\right)^4 = \left(\frac{1}{3}\right)^{2+4}.
Using a variable x helps to see this structure. Since x^2 \boldcdot {x^5} = x^7 (both sides have 7 factors that are x), if we let x = 4, we can see that 4^2 \boldcdot 4^5 = 4^7. Similarly, we could let x = \frac{2}{3} or x = 11 or any other positive value and show that these relationships still hold.
