Exponent Rules

What are Exponent Rules?

We already know how to add, subtract, and multiply. But, just as Dua Lipa has some New Rules, we have new ones of our own that we need to learn in order to simplify exponent expressions: product rules, quotient rules, and power rules..

Try out these rules in our product rule, quotient rule, and power rule calculators ⬇️.

Simplifying Exponents Calculators

Product Rules Calculator

Simplify:

DecFrac

\times

DecFrac

Step 1. Check Bases

First, we’re going to see if our bases (the blue numbers) match.

We can simplify our bases to make our job easier:

Now, even though our bases don't match right now, they're both powers of the same number. We can rewrite the inside of the parentheses for each base and then use the Power Rule like this:

Looks like our bases perfectly match, so we can move on to the next step and apply one of our product rules.

Since our bases don't match, we'll move on to the next step.

Step 2. Add Exponents

Since our bases are both

, we can add our exponents to simplify:

And that's it! Our simplified number is

Step 2. Check Exponents

Next, we’re going to see if our exponents are the same.

Since our exponents are both

, we can multiply our bases to simplify:

And that's it! Our simplified number is

Since neither our bases nor our exponents match, we can’t simplify this expression any further - our original expression

is as far as we go!

If you think that we should be able to simplify more, let us know in the feedback!

In this lesson, we’ll cover the three sets of exponent rules: Product Rules, Quotient Rules, and Power Rules. Here's a little preview of the rules we'll cover:

Recap 🧢 Exponent Rules Formulas

	Product Rule	Quotient Rule
Same Exponent	$a^{x} \times b^{x} = (a \times b)^{x}$	$\frac{a ^{x}}{b ^{x}} = (\frac{a}{b})^{x}$
Same Base	$a^{x} \times a^{y} = a^{(x + y)}$	$\frac{a ^{x}}{a ^{y}} = a^{(x - y)}$

Power Rule
$(a^{x})^{y} = a^{(x \times y)}$

Product Rules

Whenever we multiply numbers with exponents, we need to apply one of the Product Rules. The Product Rule we use depends on whether the bases or exponents of the numbers we’re multiplying match.

	Product Rule
Same Exponent	$a^{x} \times b^{x} = (a \times b)^{x}$
Same Base	$a^{x} \times a^{y} = a^{(x + y)}$

See how in both rules, the blue highlights the parts that are the same? If nothing is the same, we can't simplify. Things that are totally different can't be easily combined.

But, if there are things in common between different parts of the equation, like the bases or the exponents are the same, then we CAN combine them.

Let’s walk through an example! Do the bases or the exponents match in this equation?

4^{3} \times 5^{3} = ?

That’s right! 🎉 In this equation, the bases are different (4 and 5), but the exponents are the same (3). This means we can follow this formula, which is the Product Rule for numbers with matching exponents:

a^{x} \times b^{x} = (a \times b)^{x}

In this formula,

a

b

represent our two bases, and

x

represents the exponent. Let’s simplify the expression by plugging the numbers from our earlier equation,

4^{3} \times 5^{3} = ?

, into this formula!

4^{3} \times 5^{3} = (

\times

)

Right! Let's think about why we can do this. Let's multiply our original expression out:

4^{3} \times 5^{3} = 4 \times 4 \times 4 \times 5 \times 5 \times 5

We have three 4's and three 5's. Since order doesn't matter when we multiply, we can move our numbers around a little:

4^{3} \times 5^{3} = 4 \times 5 \times 4 \times 5 \times 4 \times 5

Now, we have the pair

4 \times 5

three times, so we can rewrite this:

4^{3} \times 5^{3} = (4 \times 5)^{3}

Merging the part that's the same (the exponent) lets us rewrite this expression, so then we can actually take it a step further and multiply out the inside:

4^{3} \times 5^{3} = (4 \times 5)^{3} = 2 0^{3}

Let’s take a look at a different example. Do the bases or the exponents match in this equation?

6^{4} \times 6^{2} = ?

That’s right! 🎉 In this equation, the bases match, so we can use the Product Rule for numbers with matching bases:

a^{x} \times a^{y} = a^{(x + y)}

Using this formula, let's try simplifying this expression:

6^{4} \times 6^{3} = ?

Like we did before, input the right values to match our Product Rule formula above:

6^{4} \times 6^{3} =

(

+

)

Great work using the product rule to simplify ! 🥳 Like we did last time, let's think about why this rule works. We can multiply out the original expression:

6^{4} \times 6^{3} = 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6

We have four 6's for the first part and three 6's for the second, giving us a total of seven 6's. But wait! That means we can just write it as:

6^{4} \times 6^{3} = 6^{(4 + 3)} = 6^{7}

Exponents: the younger siblings 👩‍👦

Think of the exponents as the little siblings of the base numbers (they are smaller 😬). When the base numbers do fancy multiplication or division, our exponents can only do the easier versions - addition and subtraction!

So, while we may be multiplying in $6^{4} \times 6^{3}$ , when we combine the 6's, we can only add the exponents:

6^{4} \times 6^{3} = 6^{(4 + 3)} = 6^{7}

So remember, if either the bases are the same or the exponents are the same, we can simplify two powers multiplied by each other using one of our Product Rules.

Simplify Your Bases

Sometimes, it may not seem like the bases of two exponential terms match — but, if you can simplify the bases at all, there may be a match hiding in plain sight!

For example,

2^{3} \times 4^{2}

can be simplified to

2^{3} \times (2^{2})^{2}

! Then, you have matching bases, and you can use the Power Rules to take care of the new exponent.

Now that we've gotten to know the Product Rules, let's move on to Quotient Rules.

Quotient Rules

Quotient Rules work pretty much the same way that the Product Rules do! Just swap out division for multiplication with the bases, and subtraction for addition with the exponents, like this:

	Quotient Rule
Same Exponent	$\frac{a ^{x}}{b ^{x}} = (\frac{a}{b})^{x}$
Same Base	$\frac{a ^{x}}{a ^{y}} = a^{(x - y)}$

Let’s practice this! Do the bases or exponents match?

\frac{1 0 ^{3}}{1 0 ^{2}} = ?

That’s right! 🎉 In this equation, the bases match, so we can use the Quotient Rule for numbers with matching bases:

\frac{a ^{x}}{a ^{y}} = a^{(x - y)}

Using this formula, let's try simplifying this expression:

\frac{1 0 ^{3}}{1 0 ^{2}} =

(

-

)

You nailed it! 🎉 Let's think about why this works. We can multiply out our original expression first:

\frac{1 0 ^{3}}{1 0 ^{2}} = \frac{1 0 \times 1 0 \times 1 0}{1 0 \times 1 0}

Now, since everything is being multiplied on the top and bottom, we can actually cancel out two of the 10's:

\frac{1 0 ^{3}}{1 0 ^{2}} = \frac{1 0 \times 1 0 \times 1 0}{1 0 \times 1 0}

Then, we're left with just one 10, which is exactly how our rule worked:

\frac{1 0 ^{3}}{1 0 ^{2}} = 1 0^{3 - 2} = 1 0^{1} = 10

Let’s try another example. Do the bases or exponents match in this equation?

\frac{8 ^{3}}{4 ^{3}} = ?

Exactly right! 🎉 In this equation, the exponents match, so we can use the Quotient Rule for numbers with matching exponents:

\frac{a ^{x}}{b ^{x}} = (\frac{a}{b})^{x}

Using this formula, let's try simplifying this expression:

\frac{8 ^{3}}{4 ^{3}} = (

\div

)

Amazing job!! 🤩 Let's think about why this one works also. We can multiply out the top and bottom of our original:

\frac{8 ^{3}}{4 ^{3}} = \frac{8 \times 8 \times 8}{4 \times 4 \times 4}

Again, since we're just doing multiplication on the top and bottom, we can rewrite this as:

\frac{8 ^{3}}{4 ^{3}} = \frac{8}{4} \times \frac{8}{4} \times \frac{8}{4}

Now, we have the fraction

\frac{8}{4}

three times, so we can rewrite this:

\frac{8 ^{3}}{4 ^{3}} = (\frac{8}{4})^{3}

Great job! Let’s keep moving: next stop, Power Rules.

Power Rules

The first Power Rule states that when you raise one term with an exponent to another power, you multiply the two exponents. It’s described by this formula:

(a^{x})^{y} = a^{(x \times y)}

Let’s simplify this expression using our formula:

(3^{2})^{5} =

(

\times

)

You nailed it! 🌟 Now, for our last step, what does this simplify to?

Nice job! 🎊 There’s one more Power Rule to learn to explain what you should do if the exponent is raised to a power:

a^{(x^{y})}

In that case, you apply the exponents one by one, like this:

2^{(2^{3})} = 2^{8} = 256

Let’s test our knowledge of these two Power Rules. Match the following terms to the Power Rule that applies to them:

Great work! Congratulations on learning about the three sets of Exponent Rules. 🎉 Celebrate with a gif!

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Exponent Rules

What are Exponent Rules?

Simplifying Exponents Calculators

Product Rules Calculator

Step 1. Check Bases

Step 2. Add Exponents

Step 2. Check Exponents

Recap 🧢 Exponent Rules Formulas

Product Rules

Exponents: the younger siblings 👩‍👦

Simplify Your Bases

Quotient Rules

Power Rules