Chapter 1 | Functions and Graphs
99
xy
3. ( b x ) y = b
4. ( ab ) x = a x b x 5. a x b x = ⎛ ⎝ a b ⎞ ⎠ x
Example 1.34 Using the Laws of Exponents
Use the laws of exponents to simplify each of the following expressions.
3
⎛ ⎝ 2 x 2/3
⎞ ⎠
a.
2
⎛ ⎝ 4 x −1/3 ⎛ ⎝ x 3 y −1
⎞ ⎠
2
⎞ ⎠
b.
−2
⎛ ⎝ xy 2
⎞ ⎠
Solution
a. We can simplify as follows: ⎛
3
3
⎞ ⎠
⎛ ⎝ x 2/3
⎞ ⎠
⎝ 2 x 2/3
2 3
2 x 2/3
x 8/3
x 2 16 x −2/3
= x
2 =
2 = 8
2 =
2 .
⎛ ⎝ 4 x −1/3
⎞ ⎠
⎛ ⎝ x −1/3
⎞ ⎠
4 2
b. We can simplify as follows: ⎛ ⎝ x 3 y −1
2
2 ⎛
2
⎞ ⎠
⎛ ⎝ x 3
⎞ ⎠
⎞ ⎠
⎝ y −1
x 6 y −2 x −2 y −4
= x 6 x 2 y −2 y 4 = x 8 y 2 .
−2 =
−2 =
⎛ ⎝ xy 2
⎞ ⎠
x −2 ⎛
⎞ ⎠
⎝ y 2
⎝ 6 x −3 y 2 ⎞ ⎠ / ⎛
⎝ 12 x −4 y 5 ⎞ ⎠ .
Use the laws of exponents to simplify ⎛
1.28
The Number e A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests P dollars in a savings account with an annual interest rate r , compounded annually. The amount of money after 1 year is A (1) = P + rP = P (1+ r ). The amount of money after 2 years is A (2) = A (1)+ rA (1) = P (1+ r )+ rP (1+ r ) = P (1+ r ) 2 . More generally, the amount after t years is A ( t ) = P (1+ r ) t .
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