# How do you simplify 4^3·4^5?

May 29, 2016

${4}^{3} \cdot {4}^{5} = {4}^{8} = 65536$

#### Explanation:

For positive integer exponents we have:

${x}^{n} = {\overbrace{x \cdot x \cdot . . \cdot x}}^{\text{n times}}$

Hence:

${x}^{m} \cdot {x}^{n} = {\overbrace{x \cdot x \cdot . . \cdot x}}^{\text{m times" * overbrace(x * x * .. * x)^"n times}}$

$= {\overbrace{x \cdot x \cdot . . \cdot x}}^{\text{m + n times}} = {x}^{m + n}$

So in our example:

${4}^{3} \cdot {4}^{5} = {4}^{3 + 5} = {4}^{8}$

$\textcolor{w h i t e}{}$
If we know our powers of $2$, then it is also helpful to use another property of exponents:

If $a , b , c > 0$ then:

${\left({a}^{b}\right)}^{c} = {a}^{b c}$

So:

${4}^{8} = {\left({2}^{2}\right)}^{8} = {2}^{2 \cdot 8} = {2}^{16} = 65536$

$\textcolor{w h i t e}{}$
Alternatively we could write:

${4}^{2} = 16$

${4}^{4} = {4}^{2} \cdot {4}^{2} = 16 \cdot 16 = 256$

${4}^{8} = {4}^{4} \cdot {4}^{4} = 256 \cdot 256 = 65536$