- \(\displaystyle 6^3 \cdot 6^7 = 6^{3+7} = 6^{10}\)
- \(2^4 \cdot 2^7 = 2^{4+7} = 2^{11}\text{.}\) Notice that \(2^{11}\) can arise other ways; for example \(2^{11} =2^{8+3} = 2^8 \cdot 2^3\)
- \(\displaystyle (4^{60})^2 = 4^{60 \times 2} = 4^{120}\)
- \(\displaystyle \frac{1}{343} = \frac{1}{7^3} = 7^{-3}\)
- \(\displaystyle 9^6 \div 9^2 = 9^{6-2} = 9^{4}\)
- \(\displaystyle \frac{3^{-4}}{3^{-5}} = 3^{-4- (-5 )} = 3^1 = 3\)
- \(\displaystyle 4^3 5^3 =(4 \times 5)^3 = 20^{3}\)
- \(\displaystyle 2^5 \cdot 8^6 = 2^5 \cdot (2^3)^6 = 2^5 \cdot 2^{18} = 2^{23}\)
- \(\displaystyle 3^2 \cdot 9^{-4} = 3^2 \cdot (3^2)^{-4} = 3^2 \cdot (3^{-8}) = 3^{-6}\)
- \(\displaystyle (35)^2 \cdot 7^9 = (5\cdot 7)^2 \cdot 7^{9} = 5^2 \cdot 7^{11}\)
- \(\displaystyle 12^3 \cdot 18^4 = (2\cdot 6)^3 \cdot (3\cdot 6)^4 = 2^3\cdot 6^3\cdot 3^4 \cdot 6^4 = 2^3\cdot 3^4 \cdot 6^7 = 3\cdot 6^3\cdot 6^7= 3\cdot 6^{10}\)
Section 1.3 Integer Exponents
For a non-negative integer \(n\text{:}\)
- The notation \(a^n\) represents \(a\) multiplied by itself \(n\) times. For example, \(5^3 = 5 \times 5 \times 5\text{.}\)
- If \(a \neq 0\text{,}\) then \(a^0 = 1\) because \(a\) multiplied by itself zero times is a product that has no terms, and a product that has no terms equals \(1\text{.}\) Note: \(0^0\) is not a number; it is an indeterminate form that will be studied in calculus.
- Multiplying exponents: When multiplying powers of the same base, add the exponents, because:\begin{equation*} a^n \times a^m = \underbrace{aa\cdots a}_{n\ \mathrm{times}} \times \underbrace{aa\cdots a}_{m\ \mathrm{times} } = \underbrace{aa\cdots a \times aa\cdots a}_{n+m\ \mathrm{times} } = a^{n+m} \end{equation*}When multiplying exponential expressions of different bases but of the same power, multiply by the bases together and raise it to the exponent, because:\begin{equation*} a^n\times b^n = \underbrace{aa\cdots a}_{n\ \mathrm{times}} \times \underbrace{bb\cdots b}_{n\ \mathrm{times} } = \underbrace{abab \cdots ab}_{n\ \mathrm{times} } = (ab)^n \end{equation*}
- Powers of Powers: When raising a power to another power, multiply the exponents, because:\begin{equation*} (a^n)^m = \underbrace{a^n a^n\cdots a^n}_{m\ \mathrm{times}}= \underbrace{aaaa\cdots aaa \times aaa\cdots aa}_{nm\ \mathrm{times} } = a^{nm} \end{equation*}
- Negative exponents are a shorthand for a power of the reciprocal, and only make sense if the base is not zero. That is, for \(n > 0\) and \(a \neq 0\text{,}\)\begin{equation*} a^{-n} = \big( a^{-1} \big)^n = \left( \frac{1}{a} \right)^n= \frac{1}{a^n}\text{.} \end{equation*}Note that \(a^{-1} = \frac{1}{a}\text{.}\)
- Dividing exponents: When dividing powers of the same base, subtract the exponents because of the meaning of negative exponents (above):\begin{equation*} a^n \div a^m = \frac{a^n}{a^m}= a^n\left(\frac{1}{a^m}\right)= a^n \times a^{-m} = a^{n-m}\text{.} \end{equation*}
- If you have different bases to different powers, sometimes you can combine by factoring. See below.
Example 1.3.1. Exponent Laws.
Exercises Practice Problems
Evaluate and put the following expressions into the lowest terms:
1.
\(z^4 \cdot z^{-5}\)
Answer.
\(z^{-1}\)
2.
\((9^3)^4\)
Answer.
\(9^{12}=3^{24}\)
3.
\((2^{-2})^{-4}\)
Answer.
\(2^8\)
4.
\(\frac{q^5}{q^2}\)
Answer.
\(q^3\)
5.
\((-4)^3 \cdot 5^3\)
Answer.
\((-20)^3 = - (20^3) = -20^3\)
