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What is exponentiation?

Many of us first encountered exponents in primary school. Today, the Fastmath team is revisiting what exponentiation means, its useful properties, and the powers that often come in handy in mental arithmetic games. Let’s get started!

Exponentiation is a mathematical operation involving two numbers, AA and NN, written as ANA^N. We call AA the base and NN the exponent. The expression ANA^N represents multiplying AA by itself NN times.

AN=A×A×⋯×A⏟N factorsA^N = \underbrace{A\times A\times\dots\times A}_{N\text{ factors}}

Examples of evaluating powers:

23=2×2×234=3×3×3×352=5×571=7\begin{aligned} 2^3 &= 2\times2\times2 \\ 3^4 &= 3\times3\times3\times3 \\ 5^2 &= 5\times5 \\ 7^1 &= 7 \end{aligned}

Laws of exponents

Exponents have several useful properties. Applying them can make calculations much faster. Here are the basic laws to know:

A1=AA0=1AM⋅AN=AM+NAMAN=AM−N(AM)N=AM⋅N(A⋅B)N=AN⋅BNA−1=1AA−N=1ANA1N=ANAMN=AMN\begin{aligned} A^1 &= A&A^0 &= 1 \\\\ A^M\cdot A^N &= A^{M+N}&\frac{A^M}{A^N} &= A^{M-N} \\\\ \left(A^M\right)^N &= A^{M\cdot N}&\left(A\cdot B\right)^N &= A^N\cdot B^N \\\\ A^{-1} &= \frac1A&A^{-N} &= \frac{1}{A^N} \\\\ A^\frac1N&=\sqrt[N]{A}&A^\frac MN&=\sqrt[N]{A^M} \end{aligned}

Frequently used powers

The Fastmath team has collected useful powers with results no greater than 1,024 for you to use.

22=423=824=1625=3226=6427=12828=25629=512210=102432=933=2734=8135=24336=729\begin{aligned} 2^2&=4 & 2^3&=8 & 2^4&=16 \\ 2^5&=32 & 2^6&=64 & 2^7&=128 \\ 2^8&=256 & 2^9&=512 & 2^{10}&=1024 \\ \\ 3^2&=9 & 3^3&=27 & 3^4&=81 \\ 3^5&=243 & 3^6&=729 \\ \end{aligned}
42=1643=6444=25645=102452=2553=12554=62562=3663=21672=4973=34382=6483=51292=8193=729102=100103=1000\begin{aligned} 4^2&=16 & 4^3&=64 \\ 4^4&=256 & 4^5&=1024 \\ \\ 5^2&=25 & 5^3&=125 \\ 5^4&=625 \\ \\ 6^2&=36 & 6^3&=216 \\ \\ 7^2&=49 & 7^3&=343 \\ \\ 8^2&=64 & 8^3&=512 \\ \\ 9^2&=81 & 9^3&=729 \\ \\ 10^2&=100 & 10^3&=1000 \\ \end{aligned}

Some different powers give the same result. Here they are grouped together:

24=1642=1626=6443=6482=6434=8192=8128=25644=256162=25629=51283=51254=625252=62536=72993=729272=729210=102445=1024322=1024\begin{aligned} 2^4&=16 & 4^2&=16 \\\\ 2^6&=64 & 4^3&=64 & 8^2&=64 \\\\ 3^4&=81 & 9^2&=81 \\\\ 2^8&=256 & 4^4&=256 & 16^2&=256 \\\\ 2^9&=512 & 8^3&=512 \\\\ 5^4&=625 & 25^2&=625 \\\\ 3^6&=729 & 9^3&=729 & 27^2&=729 \\\\ 2^{10}&=1024 & 4^5&=1024 & 32^2&=1024 \end{aligned}

We hope this refresher helps you understand exponents, remember their properties and common values, and apply them in your studies, problem solving, or mental arithmetic games. For more articles like this, follow us on Facebook!

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