1. Simplifying Expressions with Exponents

2. Simplify x6 × x5  The rules tell me to add the exponents. Some students have trouble keeping the rules straight, so just think about what exponents mean. The " x6 " means "six copies of xmultiplied together", and the " x5 " means "five copies of x multiplied together". So if I multiply those two expressions together, I will get eleven copies of x multiplied together. That is:  X^6 × x^5 = (x^6)(x^5) = (xxxxxx)(xxxxx) (6 times, and then 5 times) = xxxxxxxxxxx (11 times) = x^11  X^6 × x^5 = (x^6)(x^5) = (xxxxxx)(xxxxx) (6 times, and then 5 times) = xxxxxxxxxxx (11 times) = x^11  Thus:  X^6 × x^5 = x11

3. Simplify the following expression: Simplify the following expression:  The exponent rules tell me to subtract the exponents. But let's suppose that I've forgotten the rules again. The " 6^8 " means I have eight copies of 6 on top; the " 6^5 " means I have five copies of 6 underneath.  How many extra 6's do I have, and where are they? I have three extra 6's, and they're on top. Then:

4. Simplify the following expression:  How many extra copies of t do I have, and where are they? I have two extra copies, on top:

5. Simplify the following expression:  How many extra copies of 5 do I have, and where are they? I have six extra copies, underneath:

6. Simplify (–46x^2y^3z)^0  This is simple enough: anything to the zero power is just 1.  (–46x^2y^3z)^0 = 1

7. Simplify the following expression: Simplify the following expression:  I can cancel off the common factor of 5 in the number part of the fraction:  Now I need to look at each of the variables. How many extra of each do I have, and where are they? I have two extra a's on top. I have one extra b underneath. And I have the same number ofc's top and bottom, so they cancel off entirely. This gives me:

8. SimplifyingExpressions with Negative Exponents

9. Negative Exponents  Recall that negative exponents mean to move the base to the other side of the fraction line. For instance: negative exponentsnegative exponents  In the context of simplifying with exponents, negative exponents can create extra steps in the simplification process.

10. Simplify the following:Simplify the following:  The negative exponents tell me to move the bases, so:  Then I cancel as usual, and get:

11. Working with Exponents  When working with exponents, you're dealing with multiplication. Since order doesn't matter for multiplication, you will often find that you and a friend (or you and the teacher) have worked out the same problem with completely different steps, but have gotten the same answer in the end. This is to be expected. As long as you do each step correctly, you should get the correct answers. Don't worry if your solution doesn't look anything like your friend's; as long as you both got the right answer, you probably both did it "the right way".

12. Simplify the following expression: (–3x^–1y^2)^2  I can either take care of the squaring outside, and then simplify inside, or else I can simplify inside, and then take the square through. Either way, I'll get the same answer; to prove this, I'll show both ways.  I can either take care of the squaring outside, and then simplify inside, or else I can simplify inside, and then take the square through. Either way, I'll get the same answer; to prove this, I'll show both ways. simplifying firstsquaring first

13. Simplify the following expression: (–5x^–2y)(–2x^–3y^2)  Again, I can work either of two ways: multiply first and then handle the negative exponents, or else handle the exponents and then multiply the resulting fractions. I'll show both ways. multiplying first/doing the exponents first  Neither solution method is "better" or "worse" than the other. The way you work the problem will be a matter of taste or happenstance, so just do whatever works better for you.

14. Simplify the following expression:  The negative exponent is only on the x, not on the 2, so I only move the variable:

15. Simplify the following expression:  The "minus" on the 2 says to move the variable; the "minus" on the 6 says that the 6 is negative. Warning: These two "minus" signs mean entirely different things, and should not be confused. I have to move the variable; I should not move the 6.

16. Simplifying Expressions with Exponents: Complicated Examples Simplifying Expressions with Exponents: Complicated Examples

17. Simplify the following expression: Before I can cancel anything off, I need to simplify that top parentheses, because it has a negative exponent on it. I can't cancel off, say, the a's, because that a4 isn't really on top. I can either move the whole parentheses down, square, and then simplify, or I can take the negative-square through first.

18. Simplify the following expression:  This is a special case. The negative exponent says that whatever is on top should go underneath, and whatever is underneath should go on top. So I'll just flip the fraction (remembering to change the power from a negative to a positive), and simplify from there.  Warning: This only works if the negative exponent is on the whole fraction.

19. Simplify the following expression: flip inside, simplify, negative cube, flip, and simplify: flip inside, simplify, flip the fraction, and cube: flip the fraction, simplify inside, cube, flip the negative exponents, and simplify: flip the fraction, flip the negative exponents, simplify, and cube:

21. 6 7 8 9 10