Dividing with Exponent Rules
Dividing with Exponent Rules Math 97 Supplement 3 LEARNING OBJECTIVES 1. Simplify expressions that involve a monomial divided by a monomial.
Reducing Fractions When you simplify a fraction, you can divide out factors from the numerator and denominator. 84 For example, suppose we want to reduce . 180 One way of doing this is to list out the factors of each number, then divide out the common factors from the numerator and denominator:
84 2 2 3 7 7 180 2 2 3 3 5 15 Note that you can only divide out factors that are multiplied, and not terms that are added: 4 9 13 which doesn’t reduce. So you cannot divide out the 4’s or reduce the 9 and 12 4 12 16 because of the addition in the numerator and denominator. You should notice in this section that we will not have any addition or subtraction in the numerator and denominator of each example.
Reducing Fractions with Variables When you divide fractions with variables, you can use the same idea as dividing numbers: x4 x x x x 1 2 6 x x x x x x x x
x 7 x x x x x x x x 2 x2 5 x x x x x x 1
Notice that you can subtract the exponents to get the result in the final answer. After subtracting the exponents (large exponent – small exponent), make sure to leave the remaining variables in the same position that contained the higher exponent.
Example 1 Simplify the following expressions: a.
12 x5 8 x12
b.
12 x 5 y12 20 xy15
c.
9x2 36 x 6
Solution: 12 x5 4 3 x 5 3 7 a. 12 12 (7) 8x 4 2 x 2x
12 x 5 y12 4 3 x 5 (4) y 12 3 x 4 3 b. 20 xy15 4 5 x y 15 (3) 5y c.
9 x2 9 x 2 1 4 6 6 (4) 36 x 4 9 x 4x
Example 2 Simplify the following expressions:
8 x3 a. 7 14 x
2
27 x 8 b. 6 6x
3
Solution: 2
2
8 x3 2 4 x 3 4 42 16 a. 4 2 7 7 (4) 49 x8 14 x 2 7 x 7 x 72 x4 2
3 2 27 x8 3 9 x 8 (2) 9 x 729 x6 b. 6 6 23 8 6 x 3 2 x 3
3
3
2
KEY TAKEAWAYS When dividing variables with exponents that are factors in a fraction, subtract the exponents, leaving the remaining base and exponent in the same position (numerator or denominator)
TOPIC EXERCISES Divide and Simplify.
7 x4 y 2 12. 49 xy
14 x3 1. 2x 35 x12 2. 45 x 7 3. 4.
3
13.
5
12 x10
8x 4 x5
15.
26 x 6 x7
6 x 15x7 9x2
16x 3x 16. 7
11
40 x20
8x4 17. 6 2x
14 x 3 y 16 xy 3
8.
14 x5 2
12c 3 6. 90c 4 7.
8
4 x 9 x 14.
25m5 75m 6
9
5.
7 x 6 x
10 x 2 y 8 15 x 6 y 5
5
7 x5 18. 2 21x
3
20 x 9 19. 6 15 x
2
2
33 x y 22 x 4 y 8
5 x12 20. 2 40 x
3x5 y 11. 30 x 6 y 8
16 x 7 21. 6 40 x
4 x3 y 5 9. 18 x 3 y 2 6
10.
9
4
3
ANSWERS 1. 7x 2 2. 1 3. 3m 4. 13x 2 5. 3
6.
7 x2 7. 8 y2
8.
2 y3 9. 9 10. 1 11. 10xy 7 12. 13. 3x6 14. 15. 10x6 16. 1024 17. x10 18. 16 x 6 19. 9 20. 16 x 4 21. 625 22. 23.
4
Reducing Fractions When you simplify a fraction, you can divide out factors from the numerator and denominator. 84 For example, suppose we want to reduce . 180 One way of doing this is to list out the factors of each number, then divide out the common factors from the numerator and denominator:
84 2 2 3 7 7 180 2 2 3 3 5 15 Note that you can only divide out factors that are multiplied, and not terms that are added: 4 9 13 which doesn’t reduce. So you cannot divide out the 4’s or reduce the 9 and 12 4 12 16 because of the addition in the numerator and denominator. You should notice in this section that we will not have any addition or subtraction in the numerator and denominator of each example.
Reducing Fractions with Variables When you divide fractions with variables, you can use the same idea as dividing numbers: x4 x x x x 1 2 6 x x x x x x x x
x 7 x x x x x x x x 2 x2 5 x x x x x x 1
Notice that you can subtract the exponents to get the result in the final answer. After subtracting the exponents (large exponent – small exponent), make sure to leave the remaining variables in the same position that contained the higher exponent.
Example 1 Simplify the following expressions: a.
12 x5 8 x12
b.
12 x 5 y12 20 xy15
c.
9x2 36 x 6
Solution: 12 x5 4 3 x 5 3 7 a. 12 12 (7) 8x 4 2 x 2x
12 x 5 y12 4 3 x 5 (4) y 12 3 x 4 3 b. 20 xy15 4 5 x y 15 (3) 5y c.
9 x2 9 x 2 1 4 6 6 (4) 36 x 4 9 x 4x
Example 2 Simplify the following expressions:
8 x3 a. 7 14 x
2
27 x 8 b. 6 6x
3
Solution: 2
2
8 x3 2 4 x 3 4 42 16 a. 4 2 7 7 (4) 49 x8 14 x 2 7 x 7 x 72 x4 2
3 2 27 x8 3 9 x 8 (2) 9 x 729 x6 b. 6 6 23 8 6 x 3 2 x 3
3
3
2
KEY TAKEAWAYS When dividing variables with exponents that are factors in a fraction, subtract the exponents, leaving the remaining base and exponent in the same position (numerator or denominator)
TOPIC EXERCISES Divide and Simplify.
7 x4 y 2 12. 49 xy
14 x3 1. 2x 35 x12 2. 45 x 7 3. 4.
3
13.
5
12 x10
8x 4 x5
15.
26 x 6 x7
6 x 15x7 9x2
16x 3x 16. 7
11
40 x20
8x4 17. 6 2x
14 x 3 y 16 xy 3
8.
14 x5 2
12c 3 6. 90c 4 7.
8
4 x 9 x 14.
25m5 75m 6
9
5.
7 x 6 x
10 x 2 y 8 15 x 6 y 5
5
7 x5 18. 2 21x
3
20 x 9 19. 6 15 x
2
2
33 x y 22 x 4 y 8
5 x12 20. 2 40 x
3x5 y 11. 30 x 6 y 8
16 x 7 21. 6 40 x
4 x3 y 5 9. 18 x 3 y 2 6
10.
9
4
3
ANSWERS 1. 7x 2 2. 1 3. 3m 4. 13x 2 5. 3
6.
7 x2 7. 8 y2
8.
2 y3 9. 9 10. 1 11. 10xy 7 12. 13. 3x6 14. 15. 10x6 16. 1024 17. x10 18. 16 x 6 19. 9 20. 16 x 4 21. 625 22. 23.
4
