Properties of powers

Calculating with numbers: Calculating with powers and roots

Properties of powers

The following properties for powers can make calculations with powers easier. They follow almost immediately from the definition of powers. You can read the properties from left to right and from right to left; properties in practice be used in both ways.

Product of powers

When you multiply powers with the same base, you get a power with this base and you can add the exponents.

For base \(g\) and integers \(m\) and \(n\) we have: \[g^m\times g^n=g^{m+n}\]

Examples

\[\begin{aligned} 3^2\times 3^4&=\blue{ \overbrace{\color{black}{(3 \times 3)} }^{2\mathrm{\;times}}\color{black}{\times}\overbrace{\color{black}{(3 \times 3\times 3\times 3)} }^{4\mathrm{\;times}}} \\[0.2cm] &= \blue{ \overbrace{\color{black}{3 \times 3\times 3\times 3\times 3\times 3}}^{(2+4)\mathrm{\;times}}}\\[0.2cm] &=3^{2+4}\\[0.2cm] &=3^6\end{aligned}\]

Warning in general we have: \[g^m\times g^n\neq g^{m\times n}\] For example: \[\begin{aligned} 2^2\times 2^3&=4\times 8\\ &=32\\ &=2^5 ({}=2^{2+3})\\ &\neq 2^6 ({}=3^{2\times 3})\end{aligned}\]

Quotient of powers

When you divide powers with the same base, you get a power with this base and you can subtract the exponents from each other

For base \(g\) and integers \(m\) and \(n\) we have: \[g^m\div g^n=g^{m-n}\]

Examples

\[\begin{aligned} 3^4\div 3^2&=\blue{ \overbrace{\color{black}{(3 \times 3\times 3\times 3)} }^{4\mathrm{\;times}}\color{black}{\div}\overbrace{\color{black}{(3 \times 3)} }^{2\mathrm{\;times}}} \\[0.2cm] &= \blue{ \overbrace{\color{black}{(3 \times 3)} }^{(4-2)\mathrm{\;times}}}\\[0.2cm] &=3^{4-2}\\[0.2cm] &=3^2\end{aligned}\]

Warning In general we have: \[g^m\div g^n\neq g^{m\div n}\] For example: \[\begin{aligned} 3^6\div 3^2&=729\div 9\\ &=81\\ &=3^4 ({}=3^{6-2})\\ &\neq 3^3 ({}=3^{6\div 2})\end{aligned}\]

Power of powers

When you raise a power to a power, then you get a new power with the same base and you can multiply the exponents.

For base \(g\) and integers \(m\) and \(n\) we have: \[\left(g^m\right)^n=g^{m\times n}\]

Examples

\[\begin{aligned} \left(4^3\right)^2&=\blue{ \overbrace{\color{black}{(4^3 \times 4^3)} }^{2\mathrm{\;times}}} \\[0.2cm] &= \blue{ \overbrace{\blue{ \overbrace{\color{times}{(4 \times 4\times 4)}}^{3\mathrm{\;times}}\color{black}{\times}\overbrace{\color{black}{(4 \times 4\times 4)}}^{3\mathrm{\;times}}}}^{2\mathrm{\;maal}}}\\[0.2cm] &= \blue{ \overbrace{\color{black}{4 \times 4\times 4\times 4\times 4\times 4}}^{(3+3)\mathrm{\;times}\;=\;(3\times 2)\mathrm{\;maal}}}\\[0.2cm] &=4^{3\times 2}\\[0.2cm] &=4^{6}\end{aligned}\]

Additional example In the derivation of this property we can also use the property of the product of powers. We illustrate this with a second example: \[\begin{aligned} \left(2^3\right)^4&=\blue{ \overbrace{\color{black}{(2^3 \times 2^3\times 2^3\times 2^3)} }^{4\mathrm{\;times}}} \\[0.2cm] &= 2^{\blue{\overbrace{\color{black}{3+3+3+3}}^{4\mathrm{\;times}}}} \\[0.2cm] &=2^{3\times 4}\\[0.2cm] &=2^{12}\end{aligned}\]

Warning In general we have: \[\left(g^m\right)^n\neq g^{(m^n)}\] example \[\begin{aligned} \left(2^3\right)^2&=8^2\\ &=64\\ &=2^6 ({}=2^{3\times 2})\\ &\neq 2^9 ({}=2^{(3^2)})\end{aligned}\]

Power of a product

The power of a product is equal to the product of powers, in which the bases are the factors of the product.

Conversely, if you multiply two powers with the same exponent, you get a power with the same exponent and you can multiply the bases.

For bases \(g\) and \(h\) and integer \(n\) we have: \[(g\times h)^n = g^n\times h^n\]

Examples

\[\begin{aligned} (4\times 3)^2&=\blue{ \overbrace{\color{black}{(4 \times 3)\times (4\times 3)}}^{2\mathrm{\;times}}}\\[0.2cm] &=\blue{ \overbrace{\color{black}{(4 \times 4)}}^{2\mathrm{\;times}}}\times \blue{ \overbrace{\color{black}{(3 \times 3)}}^{2\mathrm{\;times}}}\\[0.2cm] &=4^2\times 3^2\end{aligned}\]

Power of a quotient

The power of a quotient is equal to the quotient of powers, in which the bases are the numerator and the denominator of the quotient.

Conversely, when you divide a power by another power with the same exponent, then you get a power with the same exponent and you can divide the bases.

For bases \(g\) and \(h\) and integer \(n\) we have: \[\left(g\div h\right)^n = g^n\div h^n\] or written with some stripes: \[\left(\frac{g}{h}\right)^n = \frac{g^n}{h^n}\]

Examples

\[\begin{aligned} \left(\frac{2}{3}\right)^4&=\blue{ \overbrace{\color{black}{\left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)}}^{4\mathrm{\;times}}}\\[0.2cm] &=\blue{ \overbrace{\color{black}{\frac{2 \times 2\times 2\times 2}{3\times 3\times 3\times 3}}}^{4\mathrm{\;times}}}\\[0.2cm] &=\frac{2^4}{3^4} \end{aligned}\]