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

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

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

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

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

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

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

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

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

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