Take the 2-minute tour ×
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.

I'm not sure how to simplify a fraction with a large exponent for example:

$$\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}$$

share|improve this question
3  
hint: $6 = 2*3$, and $(2*3)^{2002} = 2^{2002}*3^{2002}$. Are you happy from there? –  user29743 Aug 30 '12 at 22:50
2  
Just a nit-pick about terminology in the title: fractions cannot be solved. They can be simplified. Equations are solved. –  Old John Aug 30 '12 at 23:09
add comment

3 Answers

up vote 5 down vote accepted

To simplify a fraction with powers in the numerator and denominator a possible method is to factor each power base into prime factors. With practice it can be done directly if the bases are small numbers. $2$ and $3 $ are prime numbers. So we need only to factor $6=2\cdot 3$: $$\begin{eqnarray*} \frac{2^{2001}3^{2003}}{6^{2002}} &=&\frac{2^{2001}3^{2003}}{\left( 2\cdot 3\right) ^{2002}}=\frac{2^{2001}3^{2003}}{2^{2002}3^{2002}},\qquad (ab)^n=a^nb^n. \end{eqnarray*}$$

Now we compute the exponents by the rules $\dfrac{a^{p}}{a^{q}}=a^{p-q}$ and $ \dfrac{b^{n}}{b^{m}}=\dfrac{1}{b^{m-n}}$ $$\begin{equation*} \frac{2^{2001}3^{2003}}{2^{2002}3^{2002}}=\frac{3^{2003-2003}}{2^{2002-2001}} =\frac{3^{1}}{2^{1}}=\frac{3}{2}, \end{equation*}$$

or factor each power and divide both numerator and denominator by the common factors $$\begin{equation*} \frac{2^{2001}3^{2003}}{2^{2002}3^{2002}}=\frac{2^{2001}3\cdot 3^{2002}}{2\cdot 2^{2001}3^{2002}}=\frac{3}{2}. \end{equation*}$$

share|improve this answer
add comment

$$\displaystyle \frac{2^{2001} \cdot 3^{2003}}{6^{2002}}= \frac{2^{2001}\cdot 3^{2003}}{2^{2002} \cdot 3^{2002}}=\frac{3}{2}. $$

share|improve this answer
add comment

$$\frac{2^{2001} 3^{2003}}{6^{2002}}\\ = \frac{2^{2001} 3^{2003}}{2^{2002}3^{2002}}\\ = \frac{2^{2001}}{2^{2002}}\frac{3^{2003}}{3^{2002}}\\ = 2^{2001 - 2002}3^{2003 - 2002} \\= 2^{-1}3^1 = \frac{3}{2}$$

share|improve this answer
add comment

Your Answer

 
discard

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.