Hippocampus Algebra: Quotient of powers

Previously, we learned about the power of a power rule. Let’s take a short review.

You are given a power of a power. Find its value.

Enter the value into the text box.

Click the "Submit" button to see the answer.

Now, let’s move on with today’s lesson.

Suppose we want to divide &space; &space; &space; <EQUATION>.

Let’s rewrite the numerator as <EQUATION>.

Next, we rewrite the denominator as <EQUATION>.

We then carry out the division on the right. This gives the expression: <EQUATION>

Now, what is the term on the right, written as a power? Click the "Submit" button after selecting your answer.

Now, we have: <EQUATION>. We see that a quotient of powers equals a power.

Look at the exponents.

What can be done to the exponents of the left to result in the exponent on the right? Click the "Submit" button after selecting your answer.

We see that the exponent on the right is just the difference between the exponents in the numerator and denominator on the left: 6 – 4 = 2

This shows the quotient of powers rule.

We divide powers of the same base by subtracting exponents.

In our example, we are dividing two powers with base 5.

The value of the resulting power is 25.

Now, you try some examples.

You are given a quotient of powers. Find the exponent of the resulting power.

Enter the value into the exponent text box.

Click the "Submit" button to check your answers.

So far, we have considered only terms with positive exponents.

Here we have a term with a negative exponent: <EQUATION>

We can rewrite this negative power using our quotient rule. It is the same as <EQUATION>

Now remember that any number to the zero power is just 1.

This shows us the negative exponent rule.

We have that: <EQUATION>

Now, look at this quotient containing a negative exponent: <EQUATION>

What power term is this equal to? Click the "Submit" button after selecting your answer.

We find that: <EQUATION>

This is just another form of the negative exponent rule.