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Simplifying Expressions with Exponents - Problem 3
When simplifying a variable expression with exponents, start with the 0 and negative exponents. Remember that anytime being raised to the 0 power equals 1. This is true for variable or constants. For negative expressions, take the reciprocal of the base. Once all the 0 and negative exponents are simplified, write out the exponents. For example, write out x^3 as x x x. Next, cancel out anything that is both in the numerator and denominator. For example, if there are 3 x's in the numerator and 5 in the denominator, then 3 x's can be cancelled out from the both numerator and denominator, leaving 2 x's in the denominator. After canceling, multiply everything back together. The result will be the expression in simplest form.
Here I have an exponent problem that’s a little intimidating because there’s tons of letters, there’s tons of numbers, there’s tones of exponents. Some are positives oh my gosh, I’m like overwhelmed just looking at it. What I want to do is go through step by step and take care of the exponents one at a time.
The first thing that jumps out at me is that term where I have a negative exponent. See that a to the 0 piece? That’s the same thing as 1. Be careful, the 3 is still going to remain, but anything to the 0 power is just plain old one.
The next thing I’m going to do is go through and write out the top where this 2 will be distributed, kind of distributed. This 2 will be applied to 3 and also to b I’m also going to rewrite the bottom so I can get rid of that negative exponent. Here we go.
On top I’m going to have 3² b² times b to the fourth c, that’s just this top piece. Now I’m ready to tackle the bottom. On the bottom I have 3 to the third, and then this -1 means that this whole piece that whole chunk, the ab² moves into the top, ab². The squared stays with the b, nothing got negative anymore just this whole unit moved up to the top. I still have c to the third in the bottom. That’s the most confusing step for most students, so if I lost you rewind and start this video again.
Once I’m here you know me I like to write everything out, I’m going to go through and write out all of these letters. 3² is equal to 9 b, b 4 more bs, then a c, then an a, then 2 more bs. That’s all my letters written out. On the bottom 3×3 is 9×3 again is 27, and then I have 3 cs. Now what I would do is cancel out anything that’s the same on top and bottom, or reduce any fractions, so 9 over 27 can reduce to one third and then c on top with 3 cs on the bottom, I’m just going to have 2 cs left. Let’s go ahead and write our final answer by combining all the letters.
I’m going to do a first because I feel like going alphabetically, but it would still be correct if you wrote your b term first. So I’m just going to write my a value first because I don’t know alphabetically a comes first. It doesn’t matter what products, you can write either one first. I’ve 1 a and then I have 1, 2, 3, 4, 4, 5, 6, 7, 8, b to the eighth. On the bottom I have a 3 and then c times itself. That’s my most simplified form and I know because I don’t have any letters that show up the same on top and bottom. I don’t have any fractions with numbers to reduce and I don’t have any negative exponents.
Who would have thought that that whole old ugly enchilada simplifies to just this little kind of cute fraction. That’s the neat thing about exponents. If you write everything out, you can keep track of what’s going on and go from what used to be a really nasty fraction in towards a much more approachable fraction.