# How do you write three different expressions that can be simplified to x^6?

Jul 6, 2018

Here are some examples...

#### Explanation:

You can construct some simple ones using the laws of exponents:

${x}^{a} \cdot {x}^{b} = {x}^{a + b}$

which holds for $x > 0$ or for any $x$ if $a , b$ are non-negative integers.

For example:

${x}^{2} \cdot {x}^{4} = {x}^{2 + 4} = {x}^{6}$

You can also construct interesting expressions by taking differences of squared binomials:

$\frac{1}{2} \left({\left({x}^{6} + \frac{1}{2}\right)}^{2} - {\left({x}^{6} - \frac{1}{2}\right)}^{2}\right)$

$= \frac{1}{2} \left(\left({x}^{12} + {x}^{6} + \frac{1}{4}\right) - \left({x}^{12} - {x}^{6} + \frac{1}{4}\right)\right) = {x}^{6}$

Or you can make something really nasty by finding an algebraic expression that simplifies to $6$, e.g.

${x}^{\sqrt[3]{135 + 78 \sqrt{3}} + \sqrt[3]{135 - 78 \sqrt{3}}} = {x}^{6}$