forget how to do this

(NOTE: You should see exponents (superscripts (and not ^)) throughout this explanation. If you are not seeing superscripts, the answer is not displaying correctly.)

Hi Mathalina.

PEMDAS says you should really start with P: inside the parenthesis first, but there's not much to do there. Then E: exponents.

In this problem you have an power raised to a power. There's a rule: (x^{m})^{n} = x^{mn}. All that means is when you have a power to a power, you MULTIPLY the exponents. It's like "distributing". But you have to be really careful with the signs. And when you "distribute" that outer exponent, you must distribute to EVERYTHING inside the parenthesis, including the plain numbers (like the 2 in the numerator). Just be super careful with your signs.

So your equation becomes, after "distributing" the exponent (-5) on the top and the exponent (2) on the bottom:

( 2^{-5} * a^{-5 }* b^{10 }* c^{20 }) / (16^{2} * a^{-6} * b^{2} * c^{10} )

Now there are rules for exponents, but sometimes just simpler to see what is going all by making all the exponents positive. This is not the most direct way to solve this at this point, but it can be instructive. Remember that rule, x^{-n} = 1/x^{n}, which simply means you can flip something with an exponent from the top to the bottom (or vice versa) by changing the sign of the exponent. Let's do that now to the above equation by taking something with a negative exponent, flipping up to down, or down to up, AND changing the sign of that exponent:

( a^{6} * b^{10} * c ^{20)} / (2^{5} * 16^{2}^{ }* a^{5} * b^{2} * c^{10
})

Let's just consider the a's. You have 6 lined up in a row all multiplied together on the top, and 5 in a row all multiplied together on the bottom. All but one will cancel, with the leftover one on the top:

(a*a*a*a*a*a) / (a*a*a*a*a) = a

Likewise with the b's: 10 all multiplied together on the top, 2 on the bottom, so after cancelling you have 8 left on the top.

By cancelling in this fashion you are left with:

(a^{1} * b^{8} * c^{10} ) / (2^{5} * 16^{2} )

Now use your calculator for the numbers (using the y^{x} or the ^ button (depending on your calculator), or just multiply them out) and you get:

(a b^{8} c ^{10} ) / (8192 )

Hope that helps.

Diane

## Comments

If you aren't sure about the rule for a power raised to a power when taking a test, you can go back to basics and figure it out or check your memory.

(a

^{3})^{2}= (a*a*a)^{2 }(a*a*a)^{2 }= (a*a*a) * (a*a*a)so...

(a

^{3})^{2}= a^{6}6 = 3*2

Just think about what those exponents mean and expand a simple example formula. (Be careful not to make it

toosimple. Don't use all 1's and 2's for your plug-in numbers because it might be a special case that wouldn't apply to all possible numbers. Use numbers that aren't multiples of each other.) With a little practice, you can do a lot of simple checks like this in your head. Knowing that you have the rule right takes a lot of the stress out of test taking!Gene