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Word Problems: Equivalent Expressions

Sometimes, we need to translate a statement of equivalence back and forth between words and symbols.

For example, "Cubing a number and then dividing the result by $8$ is the same as dividing the number by $2$ and then finding its cube."

Let’s translate this into math symbols. “Cubing a number” means multiplying it $3$ times, or

$x\cdot x\cdot x=x^3$

Now divide this by $8$ :

$\frac{x^3}{8}$

On the other side, start by dividing $x$ by $2$ , and then cube the result. (Use parentheses.)

${\left(\frac{x}{2}\right)}^3$

So the translation of the full statement becomes:

$\frac{x^3}{8}={\left(\frac{x}{2}\right)}^3$

(In this case, you can check that the equation is true using the Power of a Quotient Property .)