Hippocampus Algebra: Power of powers

Previously, we learned about the product of powers rule. Let’s review it.

You are given a product of powers. Use the product of powers rule to simplify it.

Enter the values of the missing base and exponent into the text boxes.

Click the "Submit" button to see the answer.

Now, let’s continue with today’s lesson.

Here we have <EQUATION>

Remember that this is the same as <EQUATION>.

Now, let’s group the tens into <EQUATION>.

Each of these terms can be written as the exponential term <EQUATION>.

Next, let’s represent <EQUATION> by the letter <EQUATION>.

How can we write the terms on the left as a single exponential term? Click the "Submit" button after selecting your answer.

We have the left side as the exponential: <EQUATION>

But remember, we are using a to represent “ten to the power of two.” Let’s substitute back.

We now have <EQUATION>

We have a power of a power.

Look at the exponents on the left side.

How can we make them equal the value of the exponent on the right? Click the "Submit" button after selecting your answer.

If we multiply the exponents on the left, we end up with the same exponential term as that on the right: <EQUATION>

This shows the power of a power rule.

We find the power of a power by multiplying the exponents: <EQUATION>

Now, you do some examples on your own.

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.

Suppose we have the product <EQUATION>.

We can use the commutative and associative laws to rewrite the expression as: <EQUATION>

Now, we have<EQUATION>.

And we have<EQUATION>.

How can we rewrite the expression on the right as two exponential terms? Click the "Submit" button after selecting your answer.

We have the right side written as the product of powers: <EQUATION>

We also can rewrite the left side as <EQUATION>

Notice that the exponents are all the same.

This shows the power of a product rule.

The power of a product is equal to the product of the powers.

Now, you do some examples on your own.

You are given a power of a product.

Enter the values of the ‘missing’ base and exponent in the resulting product of powers.

Click the "Submit" button to see the answer.

Remember, when using the power of a product rule, the exponents remain the same.