When dividing exponents with the same base, we just subtract the exponents and keep the base the same.

Although the bases are not the same, we know that . Therefore we now have 

Now, we can add the exponents. 

When we add exponents, we try to factor to see if we can simplify it. Let's factor . We get . Remember to apply the rule of multiplying exponents which is to add the exponents and keeping the base the same. Next, we can divide  on both sides.

 We know 

 With the same base, we know that .

The first step in solving an equation like this to make the base the same on both sides of the equation. Since 5 is a factor of 125, we can rewrite the equation like this:

Using the Power of a Power Property of exponents, we get:

If the bases are the same on both sides of the equation, then the exponents must be equal, so

becomes

Solving for x:

In order to solve for the unknown variable, first change the base of the value of 25 to .

The equation  becomes:

Since the bases are now the same, we can set the powers equal to each other.

Simplify the right side by distributing the integer through the binomial.

Subtract  from both sides.

Add twelve on both sides.

Divide by fifteen on both sides and reduce the fraction.

The answer is:  

Rewrite the right side of the equation using a base of ten.

One thousand to the power of x can be rewritten using the product of exponents.

Now that the bases are equal, set the powers equal to each other.

Subtract  from both sides.

Simplify both sides.

The answer is:  

In order to solve this equation, we will need to convert the nine on the right side to a base of three.

Rewrite the equation.

Set the powers equal to each other since the bases are common.

Distribute the two across the binomial.

Subtract  from both sides.

The answer is:  

In order to determine the value of x, we will need to convert the base of one-tenth to the base of ten.

Rewrite the fraction as a base of ten.

Rewrite the right side of the equation using the new term.

According to the product rule for exponents, we can set the powers equal since the bases are similar.

Divide by negative one on both sides.

Subtract  on both sides.  The equation will become:

Divide by negative two on both sides.

The answer is:  

In order to solve this, we will need to rewrite the right side as the similar base to the left side of the equation.

Rewrite the right side.

With similar bases, the powers can be set equal to each other.

Subtract  on both sides.

Add six on both sides.

Divide by two on both sides.

The answer is:  

To solve this equation, we will need to convert the 100 into base ten.

Rewrite the number using this base.

Now that the bases are similar, the exponents can be set equal to each other.

Simplify this equation.

Subtract  on both sides.

The answer is: