# Thread: Simply Fraction with negative exponets -Confused

1. ## Simply Fraction with negative exponets -Confused

$\displaystyle (\frac{2s^{-1}t^3}{6s^2t^{-4}})^{-3}$

Which i reduced to ...
$\displaystyle \frac{27t^3}{s^3}$

My work ...
$\displaystyle \frac{2^{-3}s^3t^{-9}}{6^{-3}s^{-6}t^{12}}$

$\displaystyle \frac{1}{8}*216=\frac{216}{8}=27$

$\displaystyle s^3*\frac{1}{s^6}=\frac{s^3}{s^6}=s^{-3}=\frac{1}{s^3}$

$\displaystyle \frac{1}{t^9}*t^{12}=\frac{t^{12}}{t^9}=t^3$

Paul

2. Originally Posted by pkraus
$\displaystyle (\frac{2s^{-1}t^3}{6s^2t^{-4}})^{-3}$

Which i reduced to ...
$\displaystyle \frac{27t^3}{s^3}$

My work ...
$\displaystyle \frac{2^{-3}s^3t^{-9}}{6^{-3}s^{-6}t^{12}}$

$\displaystyle \frac{1}{8}*216=\frac{216}{8}=27$

$\displaystyle s^3*\frac{1}{s^6}=\frac{s^3}{s^6}=s^{-3}=\frac{1}{s^3}$

$\displaystyle \frac{1}{t^9}*t^{12}=\frac{t^{12}}{t^9}=t^3$

Paul
Personally I would solve it by flipping the fraction and making the power +3

$\displaystyle (\frac{2s^{-1}t^3}{6s^2t^{-4}})^{-3} = (\frac{6s^2t^{-4}}{2s^{-1}t^3})^3$

$\displaystyle = \frac{6^3s^6t^{-12}}{2^3s^{-3}t^9} = \frac{6^3}{2^3} \times \frac{3s^6}{s^{-3}} \times \frac{t^{-12}}{t^9} = 27s^9t^{-21}$

For non-zero a:
$\displaystyle a^{-n} = \frac{1}{a^n}$

$\displaystyle a^na^m = a^{m+n}$

$\displaystyle \frac{a^n}{a^m} = a^{n-m}$

In your case with s you've forgot to put the minus sign on the power in the denominator (see my red text)
$\displaystyle s^3*\frac{1}{s^{{\color{red}-}6}}=\frac{s^3}{s^{{\color{red}-}6}}=s^{3-{\color{red}(-6)}}=s^9$

For t:

From your original (and correct) expansion you got
$\displaystyle \frac{t^{-9}}{t^{12}}$

However, in the next step you added the t^12 rather than subtracted it. Since t^12 is in the denominator it needs to have a negative power when multiplied (see the first law above)

$\displaystyle \frac{1}{t^9}*t^{{\color{red}-}12}=\frac{t^{{\color{red}-}12}}{t^9}=t^{-21}$

3. Originally Posted by pkraus
$\displaystyle (\frac{2s^{-1}t^3}{6s^2t^{-4}})^{-3}$

Which i reduced to ...
$\displaystyle \frac{27t^3}{s^3}$

My work ...
$\displaystyle \frac{2^{-3}s^3t^{-9}}{6^{-3}s^{-6}t^{12}}$

$\displaystyle \frac{1}{8}*216=\frac{216}{8}=27$

$\displaystyle s^3*\frac{1}{s^6}=\frac{s^3}{s^6}=s^{-3}=\frac{1}{s^3}$

$\displaystyle \frac{1}{t^9}*t^{12}=\frac{t^{12}}{t^9}=t^3$

Paul
Hi Paul,

Here's another take on it.

Here's what I'd do with that. First invert the fraction so that the group will have a positive exponent.

$\displaystyle \left(\frac{2s^{-1}t^3}{6s^2t^{-4}}\right)^{-3}=\left(\frac{6s^2t^{-4}}{2s^{-1}t^3}\right)^3$

Next, simplify the group.

$\displaystyle \left(\frac{3s^3}{t^7}\right)^3$

Finally, bring it home.

$\displaystyle \frac{27s^9}{t^{21}}$