# Power of Powers: Simplifying Exponential Expressions

## Math Building Blocks

I love Legos. There are so many options! Different colors, sizes, shapes, kits - the possibilities are endless. One day you can build a town, the next day a scene from a movie and the next day you can decide to just pile all the blocks into one free-form structure with no real purpose.

Even though there might not be as much creativity in mathematics as there is in playing with Legos, we end up doing a different kind of building when dealing with numbers and variables. And exponents even have that look of building and expanding. They always remind me of working with blocks.

## The Base of an Exponential Term

The ground floor of any exponential term is called the base.

In this term, the base is the number 2.

The ground floor of an exponent can be any number or variable.

## The Exponent

The next floor of an exponential term is the exponent - the 3 in this case.

The **exponent** tells you how many times to multiply a number to itself.

## Raising a Power to a Power

As with Legos, there are times when the exponential term has a third floor. This is called raising a power to a power and looks like this:

These exponential terms can be simplified by writing them out like this, which can then be written as:

which simplifies to 5^6 because there are six 5s being multiplied together.

If we look back to the original problem, we see that by multiplying the two exponents together, we also get 6.

2*3 = 6

So we can assume from this that to simplify a term with a power raised to a power, you just need to multiply the exponents together. But let's try another example.

Simplify:

Simplifying the long way gives us:

which equals a whole bunch of *x*s. When you add them up, you see it's *x*^15.

If we multiply the exponents we get:

3*5 = 15

so

Again we see that the exponents can be simplified by multiplying the two exponents together.

This rule also applies when there are negative exponents.

Simplify:

By simply multiplying the exponents together we see that the answer is:

(2)*(-4) = -8

so

Let's work it out the long way as proof.

As you can see when you count up all the *b*s, we get *b*^-8 as the answer, proving that the multiplication method works.

## Lesson Summary

Exponents can build upon each other just like Lego blocks can build upon each other. When you are simplifying exponential terms raised to another exponent, the way to simplify them is to multiply the exponents together. This will work with positive and negative exponents and can be shown by working out each exponent the long way.

## Learning Outcome

After watching this lesson, you should be able to interpret the rule for raising a power to a power and give examples to solve these expressions.

To unlock this lesson you must be a Study.com Member.

Create your account

#### Power Of Powers—Additional Work

At first glance, exponential expressions can appear quite intimidating and difficult to understand. However, there are three simple rules that can make tackling exponential expressions much simpler.

**The Product Rule**

The product rule is used to multiply exponential expressions with like bases. When an expression has exponential terms with like bases, you can simply add the exponents together. Let's look at an example.

{eq}(x^{3})(x^{4}) {/eq}

{eq}(x^{3+4}) {/eq}

{eq}(x^{3})(x^{4})=(x^{7}) {/eq}

If the coefficients differ between the exponential terms, first multiply the coefficients together. Then, add together the exponents with a common base. Let's look at an example.

{eq}4a^{3}\cdot5a^{2} {/eq}

{eq}20\cdot a^{3+2} {/eq}

{eq}20\cdot a^{5} {/eq}

{eq}4a^{3}\cdot5a^{2}=20a^{5} {/eq}

**The Power Rule**

The power rule applies to exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. Let's look at an example.

{eq}(5^{2})^{4} {/eq}

{eq}5^{2\cdot 4} {/eq}

{eq}5^{8}=390,625 {/eq}

**The Quotient Rule**

The quotient rule for exponents applies when exponential expressions contain division. When you divide two numbers in exponential form with the same base, you can subtract the exponent in the denominator from the exponent in the numerator. Let's look at an example.

{eq}\frac{x^{5}}{x^{2}} {/eq}

{eq}x^{5-2} {/eq}

{eq}x^{3} {/eq}

When dividing terms in an exponential expression that also contain coefficients, divide the coefficients first and then divide the variable powers with the same base by subtracting the exponents. Let's look at an example.

{eq}\frac{15x^4}{5x^2} {/eq}

{eq}(\frac{15}{5})(\frac{x^4}{x^2}) {/eq}

{eq}3(x^{4-2}) {/eq}

{eq}3x^{2} {/eq}

**Apply All 3 Rules Together**

Now that we've learned how to use the power, product, and quotient rules, let's use all three rules together in an example. Remember, when simplifying an exponential expression, be sure to follow the order of operations. Any easy way to remember the order of operations is the acronym **PEMDAS**:

**P**arantheses

**E**xponents

**M**ultiplication

**D**ivision

**A**ddition

**S**ubtraction

Let's work through an example together.

{eq}\frac{x^2(x^5)^3}{8x^8} {/eq}

First, use the power rule to multiply the exponents.

{eq}\frac{x^2(x^{15})}{8x^8} {/eq}

Second, use the product rule to multiply to add together exponents with the same base.

{eq}\frac{x^{17}}{8x^8} {/eq}

Third, use the quotient rule to subtract the denominator's exponent from the numerator's exponent.

{eq}\frac{x^{9}}{8} {/eq}

Here's the full equation reduced to its most simple form.

#### {eq}\frac{x^2(x^5)^3}{8x^8}=\frac{x^{9}}{8} {/eq}

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

Back