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### Find X

```
Date: 01/22/2001 at 15:44:12
From: Crystal Price
Subject: Find x

How do you do a problem like this?

Use your knowledge of powers and the properties of exponents to find
the value of x in the equation 8^x + 2 = 2.
```

```
From: Doctor Ian
Subject: Re: Find x

Hi Crystal,

Let's take a look at that equation:

8^x + 2 = 2

As usual, we want to combine as many numbers as we can, so we can
subtract 2 from each side of the equation:

8^x + 2 - 2 = 2 - 2

8^x = 0

Now we have a problem. Recall that 8^1 = 8, and 8^0 = 1, and 8^(1/n)
is some root of 8, so it's always greater than zero.

The problem is that there is NO way that you can raise a positive
number to any exponent to get zero. So this leads me to believe that
you really meant to say

8^(x+2) = 2

When you're using '^' to write exponents that involve more than just a
number, you _need_ to use parentheses.  That is,

x+2
8     = 8^(x+2)

8^x+2 = (8^x) + 2

Anyway, let's look at

8^(x+2) = 2

It turns out that 2 is the cube root of 8; that is,

2^3 = 8

2 = 8^(1/3)

So we can use this information:

8^(x+2) = 2

= 8^(1/3)

This tells us that

x + 2 = 1/3

Why? Just as you can add or subtract the same thing to/from both sides of
an equation, or multiply both sides by the same thing, you can also
raise both sides of an equation to the same exponent - or, if both
sides are the same base raised to an exponent, you can drop the base.
All that's saying is that if you have

a^b = a^c

then you know that b = c.  Does that make sense?

Anyway, if you solve

x + 2 = 1/3

you'll get the value of x that makes

8^(x+2) = 2

a true statement.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra
Middle School Exponents

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