Amy's Reflection #5

Exponents:

1. b^x * b^y = b^x + y....example: 2^3 * 2^5 = 2^8

2. b^x/b^y = b^x - y....example: 5^7/5^4 = 5^3

3. (ab)^x = a^xb^x....example: (3 * 7)^3 = 3^3 * 7^3

4. (a/b)^x = a^x/b^x....example: (3/5)^3 = 3^3/5^3

5. (b^x)^y = b^xy....example: (2^2)^3 = 2^6

6. b^x/y = y^√b^x....examples: 5^3/4 = 3^√5^3

7. to solve for exponents:

• write as the same base
• set exponents equal
• then solve for x

here are some examples:

(a). 5^3x = 5^7x - 2

In this first part we have the same base on both exponentials so there really isn’t much to do other than to set the two exponents equal to each other and solve for x.

3x = 7x - 2

2 = 4x

x = 1/2

So, if we were to plug x = 1/2 into the equation then we would get the same number on both sides of the equal sign.

(b). 4^t^2 = 4^6 - t

t^2 = 6 - t

t^2 - t - 6 = 0

(t - 2) (t + 3) = 0

t = -3, t = 2

In this case we get two solutions to the equation. That is perfectly acceptable so don’t worry about it when it happens.

(c). 3^z = 9^z + 5

Now, in this case we don’t have the same base so we can’t just set exponents equal. However, with a little manipulation of the right side we can get the same base on both exponents. To do this all we need to notice is that 9 = 3^2. Here’s what we get when we use this fact:

3^z = (3^2)^z + 5

Now, we still can’t just set exponents equal since the right side now has two exponents.

3^z = 3^2(z + 5)

We now have the same base and a single exponent on each base so we now set the exponents equal. Doing this gives us....

z = 2(z + 5)

z = 2z + 10

-10 = z

...a solution of z = -10.

i hope this wasn't too confusing and i was actually able to help someone...