Both positive and negative exponents are also referred to as ‘powers’ or numbers that the base number is ‘raised to the power of’. To solve an equation with a negative exponent, you must first make it positive.
To convert a negative exponent, create a fraction with the number 1 as the numerator (top number) and the base number as the denominator (bottom number). Raise the base number to the power of the same exponent, but make it positive. 3−3,5−2,{\displaystyle 3^{-3},5^{-2},} and 7−4{\displaystyle 7^{-4}} are now 1(33),1(52),{\displaystyle {\frac {1}{(3^{3})}},{\frac {1}{(5^{2})}},} and 1(74){\displaystyle {\frac {1}{(7^{4})}}}. This process is known as the negative exponent rule.
2x−1{\displaystyle 2x^{-1}} can be written as 2x−11{\displaystyle {\frac {2x^{-1}}{1}}} which can then be simplified to 2(1x1){\displaystyle {\frac {2}{({1x}^{1})}}} 21x1{\displaystyle {\frac {2}{1x^{1}}}} can then be simplified to 2x{\displaystyle {\frac {2}{x}}} In this case, only ‘x’ became the denominator because it had the exponent.
To simplify a fractional negative exponent, you must first convert to a fraction. If your starting base number is 16−1/2{\displaystyle 16^{-1/2}}, start by converting it to a fraction where the exponent becomes positive when the base number is switched to the denominator. 16−1/2{\displaystyle 16^{-1/2}} will become 1161/2{\displaystyle {\frac {1}{16^{1/2}}}} 1161/2{\displaystyle {\frac {1}{16^{1/2}}}} is equal to 1162{\displaystyle {\frac {1}{\sqrt[{2}]{16}}}} 1162{\displaystyle {\frac {1}{\sqrt[{2}]{16}}}} is equal to 14{\displaystyle {\frac {1}{4}}}.
When an exponent is negative and a base number is positive, the expression must be converted into a fraction to make the exponent positive For example, 6−2=162{\displaystyle 6^{-2}={\frac {1}{6^{2}}}} When an exponent is positive and a base number is negative, the base number will be multiplied by itself however many times the exponent shows us it should be. For example, −55=−5∗−5∗−5∗−5∗−5=−3125. {\displaystyle -5^{5}=-5*-5*-5*-5*-5=-3125. }
Remember to put negative exponent values in parentheses: 4E(−6){\displaystyle 4E(-6)} Solving exponential equations on a calculator will allow you to find answers more quickly without converting them into fractions.
4−1/4∗4−1/4{\displaystyle 4^{-1/4}*4^{-1/4}} can be simplified to 4−1/2{\displaystyle 4^{-1/2}} You can further simplify 4−1/2{\displaystyle 4^{-1/2}} into 14−1/2{\displaystyle {\frac {1}{4^{-1/2}}}} 14−1/2{\displaystyle {\frac {1}{4^{-1/2}}}} becomes 142{\displaystyle {\frac {1}{\sqrt[{2}]{4}}}} which is equal to 12{\displaystyle {\frac {1}{2}}}
Because the exponent is negative, the subtraction will cancel out the second negative and make the exponent positive. The exponents in 2−72−2{\displaystyle {\frac {2^{-7}}{2^{-2}}}} will subtract as (−7)−(−2){\displaystyle (-7)-(-2)} or (−7)+2{\displaystyle (-7)+2} The equation will simplify to 2−5{\displaystyle 2^{-5}} or 125{\displaystyle {\frac {1}{2^{5}}}}
7−6∗8−6{\displaystyle 7^{-6}*8^{-6}} will become 56−6{\displaystyle 56^{-6}} 5−1/6∗20−1/6{\displaystyle 5^{-1/6}*20^{-1/6}} will become 100−1/6{\displaystyle 100^{-1/6}}
16−1/4+4−2=1164+1(42){\displaystyle 16^{-1/4}+4^{-2}={\frac {1}{\sqrt[{4}]{16}}}+{\frac {1}{(4^{2})}}} 1164+1(42)=12+116{\displaystyle {\frac {1}{\sqrt[{4}]{16}}}+{\frac {1}{(4^{2})}}={\frac {1}{2}}+{\frac {1}{16}}} 12+116=816+116{\displaystyle {\frac {1}{2}}+{\frac {1}{16}}={\frac {8}{16}}+{\frac {1}{16}}} 816+116=916{\displaystyle {\frac {8}{16}}+{\frac {1}{16}}={\frac {9}{16}}}
