Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to simplify this fraction : $ \dfrac{(3^2)(5^4)}{15^3} $

The answer is : $ \dfrac{5}{3} $

I am trying to do the following: $ \dfrac{3^2}{15^3} \cdot \dfrac{5^4}{15^3} $ so ... $ \dfrac{1^{-3}}{3} \cdot \dfrac{1^1}{3} $

But that's not giving me the right answer, what am I doing wrong, please ?

Thanks !

share|cite|improve this question
You might want to factor the $15$ and distribute the exponents properly... – J. M. Sep 12 '11 at 8:08
up vote 7 down vote accepted

For starters, $\displaystyle\frac{3^2}{15^3} \cdot \frac{5^4}{15^3} = \frac{(3^2)(5^4)}{(15^3)(15^3)} = \frac{(3^2)(5^4)}{15^6}$, not $\displaystyle\frac{(3^2)(5^4)}{15^3}$, so your first step isn’t right. Next, $\displaystyle\frac{3^2}{15^3}$ isn’t $\displaystyle\frac{1^{-3}}{3}$, and $\displaystyle\frac{5^4}{15^3}$ isn’t $\displaystyle\frac{1^1}{3}$; these mistakes show that you have some fundamental misconception, but I’m not sure exactly what it is.

Go back to the beginning and write out the fraction without any exponents: $$\begin{align*} \frac{(3^2)(5^4)}{15^3} &= \frac{3\cdot 3\cdot 5\cdot 5\cdot 5\cdot 5}{15\cdot 15\cdot 15}\\&=\frac{3\cdot 3\cdot 5\cdot 5\cdot 5\cdot 5}{(3\cdot 5)(3\cdot 5)(3\cdot 5)}\\&=\frac{3\cdot 3\cdot 5\cdot 5\cdot 5\cdot 5}{3\cdot 3\cdot 3\cdot 5\cdot 5\cdot 5}\\&=\frac33\cdot\frac33\cdot\frac53\cdot\frac55\cdot\frac55\cdot\frac55\\&=1\cdot 1\cdot\frac53\cdot 1\cdot 1\cdot1\\&=\frac53. \end{align*}$$ When you cancel, you’re really just getting rid of factors that are equal to $1$.

I suspect that you’re actually supposed to be learning to manipulate exponents at this point, but that manipulation is just a shortcut for what I did above. You use the law that $(ab)^n = a^nb^n$ to rewrite the denominator, $15^3$, as $(3\cdot 5)^3 = 3^3 \cdot 5^3$, making your fraction $\displaystyle\frac{3^2 \cdot 5^4}{3^3 \cdot 5^3}$; if you write that out in full, you have the fraction on the third line of the displayed expressions above. Then you split it, correctly this time, as $\displaystyle\frac{3^2}{3^3}\cdot\frac{5^4}{5^3}$. Now, finally, you can use the rule that $\displaystyle\frac{a^n}{a^m}=a^{n-m}$ twice to get $3^{2-3} \cdot 5^{4-3} = 3^{-1} \cdot 5^1 = \displaystyle\frac53$.

share|cite|improve this answer

Here it is explicitly. Notice $15^3=(3\cdot 5)^3=3^3\cdot 5^3$. So $$ \dfrac{(3^2)(5^4)}{15^3}=\frac{3^2\cdot 5^4}{3^3\cdot 5^3}=\frac{5}{3}. $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.