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Simplifying Expressions with Exponents - Problem 2

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

In a variable expression with negative exponents and multiple operations, complete the operations step by step. Keep in my order of operations. Start with the exponents. To simplify a negative exponent, take the reciprocal of the term that is being raised by the exponent. Next, start simplifying by writing out all the exponents. For example, write 2x^3 as 2 x x x. Do this to every exponent. Cancel out any variables or constants that are the same in both the numerator and denominator. After canceling, multiply everything together. The result will the the expression in simplest form.

This intimidating exponent problem can be done if I do it step by step. Not only do I have 2 fractions, but they’re being multiplied and I have negative exponents oh my goodness I’m a little scared, but we can do it.

First thing I’m going to do is take care of these negative exponents. Negative exponent means whatever was in the top is going to be written in the bottom with the positive exponent. For example 3 to the -1 is going to become 3 to the 1 in the denominator. X that started out in the bottom now has a negative exponent it’s going to jump up to the top. That’s what happened in the first fraction.

The second fraction, all of this stuff is now going to be in the bottom of the fraction squared. It used to be a negative 2, now it’s going to be a positive 2. That whole chunk, (2x)². Now there’s y to the third that used to be in the bottom is now going to become y to the third to the positive 2 in top. That was hard because I had to use everything that used to be in the bottom into the top, and I had to apply my new squared.

Once I have that set up, I’m going to go through, and take care of these squared. This first fraction will stay the same, this second fraction I’m going to write out as y to the third times y to the third again, that’s what y to the third squared is. On the bottom I’ll have 2x² times 2x² because it’s timed itself.

My next step is going to be to write pout all of these Xs and Ys and then cancel out things that were the same on top and bottom. So on top all the way across I have x and then y, y, y, y, y, y again. On the bottom I have 3 times 2 times 2 which is 3 times 2 is 6 times 2 is 12 x, x, x, x again. Great now I have everything all written out, the last thing I’m going to do is cross out anything that shows up on top and bottom like that oops, not both of those Xs, just that first guy is going to get crossed out. There we go. And then I can write my simplified fractional answer.

On top I have 1, 2, 3, 4, 5 Ys no 1, 2, 3, 4, 5, 6 y, don’t write this. It’s not 6×y, it’s y times itself 6 times. Y to the 6 on top. On the bottom I have 12, one to those Xs got cancelled out, so then I’m left with x to the third. That’s my final answer and I know it’s my final answer because I don’t have any fractional integers that can be reduced. I also don’t have any fractional integers mainly like number on top of number that I can reduce. I also have different letters in the top and bottom so I know those guys can’t be simplified any further.

When I approach exponents that have negative exponents or problems that have negative exponents, I take care of the negative exponent first, then I write out everything including all the letters and cancel things that are the same on top and bottom.

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