# How to multiply numbers with exponents

Below is List of Rules for Exponents and an example or two of using each rule:

 Zero-Exponent Rule: a0 = 1, this says that anything raised to the zero power is 1. Power Rule (Powers to Powers): (am)n = amn, this says that to raise a power to a power you need to multiply the exponents. There are several other rules that go along with the power rule, such as the product-to-powers rule and the quotient-to-powers rule. Negative Exponent Rule: , this says that negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Only move the negative exponents. Product Rule: am ∙ an = am + n, this says that to multiply two exponents with the same base, you keep the base and add the powers. Quotient Rule: , this says that to divide two exponents with the same base, you keep the base and subtract the powers. This is similar to reducing fractions; when you subtract the powers put the answer in the numerator or denominator depending on where the higher power was located. If the higher power is in the denominator, put the difference in the denominator and vice versa, this will help avoid negative exponents.

Now that we have reviewed the rules for exponents, here are the steps required for simplifying exponential expressions (notice that we apply the rules in the same order the rule were written above):

 Step 1: Apply the Zero-Exponent Rule. Change anything raised to the zero power into a 1. Step 2: Apply the Power Rule. Multiply (or distribute) the exponent outside the parenthesis with every exponent inside the parenthesis, remember that if there is no exponent shown, then the exponent is 1. Step 3: Apply the Negative Exponent Rule. Negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Only move the negative exponents. Note that the order in which things are moved does not matter. Step 4: Apply the Product Rule. To multiply two exponents with the same base, you keep the base and add the powers. Step 5: Apply the Quotient Rule. This is similar to reducing fractions; when you subtract the powers put the answer in the numerator or denominator depending on where the higher power was located. If the higher power is in the denominator, put the difference in the denominator and vice versa, this will help avoid negative exponents and a repeat of step 3. Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions.

Example 1 – Simplify:

 Step 1: Apply the Zero-Exponent Rule. In this case, there are no zero powers. Step 2: Apply the Power Rule. Step 3: Apply the Negative Exponent Rule. Move every negative exponent in the numerator to the denominator and vice versa. Step 4: Apply the Product Rule. Step 5: Apply the Quotient Rule. In this case, the x’s ended up in the denominator because there were 10 more x’s in the denominator. Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions. In this case, the fraction does not reduce.

Example 2 –Simplify:

 Step 1: Apply the Zero-Exponent Rule. Step 2: Apply the Power Rule. Step 3: Apply the Negative Exponent Rule. Move every negative exponent in the numerator to the denominator and vice versa. Step 4: Apply the Product Rule. In this case, the product rule does not apply. Step 5: Apply the Quotient Rule. In this case, the quotient rule does not apply. Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions. In this case, the fraction does not reduce.

Example 3 –Simplify:

 Step 1: Apply the Zero-Exponent Rule. In this case, there are no zero powers. Step 2: Apply the Power Rule. Step 3: Apply the Negative Exponent Rule. Move every negative exponent in the numerator to the denominator and vice versa. Step 4: Apply the Product Rule. In this case, the product rule does not apply. Step 5: Apply the Quotient Rule. In this case, the x’s ended up in the numerator and the y’s ended up in the denominator. Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions. In this case, the fraction does not reduce.

Example 4 –Simplify: Step 1: Apply the Zero-Exponent Rule. In this case, after applying the zero-exponent rule and multiplying by 1, that term is essentially gone. Step 2: Apply the Power Rule. In this case, I kept the –2 in parentheses because I did not want to lose the negative sign. Step 3: Apply the Negative Exponent Rule. Move every negative exponent in the numerator to the denominator and vice versa. Step 4: Apply the Product Rule. Step 5: Apply the Quotient Rule. In this case, the x’s ended up in the denominator. Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions. In this case, the fraction does reduce.

Example 5 –Simplify: Step 1: Apply the Zero-Exponent Rule. In this case, there are no zero powers. Step 2: Apply the Power Rule. Step 3: Apply the Negative Exponent Rule. Move every negative exponent in the numerator to the denominator and vice versa. Step 4: Apply the Product Rule. In this case, we can apply the rule to the x’s and y’s in the numerator. Step 5: Apply the Quotient Rule. In this case, the x’s ended up in the numerator and the y’s ended up canceling out. Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions. In this case, the numbers in the numerator get multiplied together and then the fraction gets reduce. ## Multiplying Exponents Explained — Mashup Math

•  How can you multiply powers (or exponents) with the same base?
• Note that the following method for multiplying powers works when the base is either a number or a variable (the following lesson guide will show examples of both)

Let’s start with a simple example: what is 3^3 times by 3^2?

*Notice that each term has the same base, which, in this case is 3. Start by rewriting each term in expanded form as follows (you won’t have to do this every time, but we’ll do it now to help you understand the rule, which we’ll get to later.) Since we have 3 being multiplied by itself 5 times ( 3 x 3 x 3 x 3 x 3 ), we can say that the expanded expression is equal to 3^5 And we can conclude that: 3^3 x 3^2 = 3^5

But Why? Do you notice a relationship between the exponents?

• Did you notice a relationship between all of the exponents in the example above?
• Notice that 3^2 multiplied by 3^3 equals 3^5. Also notice that 2 + 3 = 5
• This relationship applies to multiply exponents with the same base whether the base is a number or a variable:
• Whenever you multiply two or more exponents with the same base, you can simplify by adding the value of the exponents: Here are a few examples applying the multiplying exponents rule:

## Simplify by Using the Product, Quotient, and Power Rules

• Simplify by Using the Product, Quotient, and Power Rules
• Learning Objective(s)
• ·         Use the product rule to multiply exponential expressions with like bases.
• ·         Use the power rule to raise powers to powers.
• ·         Use the quotient rule to divide exponential expressions with like bases.
• ·         Simplify expressions using a combination of the properties.

Exponential notation was developed to write repeated multiplication more efficiently. There are times when it is easier to leave the expressions in exponential notation when multiplying or dividing. Let’s look at rules that will allow you to do this.

The Product Rule for Exponents

Recall that exponents are a way of representing repeated multiplication. For example, the notation 54 can be expanded and written as 5 • 5 • 5 • 5, or 625. And don’t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.

What happens if you multiply two numbers in exponential form with the same base? Consider the expression (23)(24). Expanding each exponent, this can be rewritten as (2 • 2 • 2) (2 • 2 • 2 • 2) or  2 • 2 • 2 • 2 • 2 • 2 • 2. In exponential form, you would write the product as 27. Notice, 7 is the sum of the original two exponents, 3 and 4.

What about (x2)(x6)? This can be written as (x • x)(x • x • x • x • x • x) = x • x • x • x • x • x • x • x  or x8. And, once again, 8 is the sum of the original two exponents.

 The Product Rule for Exponents For any number x and any integers a and b, (xa)(xb) = xa+b.

To multiply exponential terms with the same base, simply add the exponents.

 Example Problem Simplify. (a3)(a7) (a3)(a7) The base of both exponents is a, so the product rule applies. a3+7 Add the exponents with a common base. (a3)(a7) = a10

When multiplying more complicated terms, multiply the coefficients and then multiply the variables.

 Example Simplify. 5a4 · 7a6 35 · a4 · a6 Multiply the coefficients. 35 · a4+6 The base of both exponents is a, so the product rule applies. Add the exponents. 35 · a10 Add the exponents with a common base. Answer 5a4 · 7a6 = 35a10
 Simplify the expression, keeping the answer in exponential notation. (4x5)( 2x8) A) 8x5 • x8 B) 6x13 C) 8x13 D) 8x40 Show/Hide Answer A) 8x5 • x8 Incorrect. 8x5• x8  is equivalent to (4x5)(2x8), but it still is not in simplest form. Simplify x5•x8 by using the Product Rule to add exponents. The correct answer is 8x13. B) 6x13 Incorrect. 6x13 is not equivalent to (4x5)(2x8). In this incorrect response, the correct exponents were added, but the coefficients were also added together. They should have been multiplied. The correct answer is 8x13. C) 8x13 Correct. 8x13  is equivalent to (4x5)(2x8). Multiply the coefficients (4 • 2) and apply the Product Rule to add the exponents of the variables (in this case x) that are the same. D) 8x40 Incorrect. 8x40 is not equivalent to (4x5)(2x8). Do not multiply the coefficients and the exponents. Remember, using the Product Rule add the exponents when the bases are the same. The correct answer is 8x13.

The Power Rule for Exponents

Let’s simplify (52)4. In this case, the base is 52 and the exponent is 4, so you multiply 52 four times: (52)4  = 52 • 52 • 52 • 52 = 58 (using the Product Rule – add the exponents).

(52)4  is a power of a power. It is the fourth power of 5 to the second power. And we saw above that the answer is 58. Notice that the new exponent is the same as the product of the original exponents: 2 • 4 = 8.

So, (52)4  = 52 • 4  =  58 (which equals 390,625, if you do the multiplication).

Likewise, (x4)3 = x4 3 = x12.

This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, (23)5 = 215.

 The Power Rule for Exponents For any positive number x and integers a and b: (xa)b= xa· b.
 Example Problem Simplify. 6(c4)2 6(c4)2 Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses. Answer 6(c4)2 = 6c8
 Example Problem Simplify. a2(a5)3 Raise a5 to the power of 3 by multiplying the exponents together (the Power Rule). Since the exponents share the same base, a, they can be combined (the Product Rule). Answer
 Simplify: A) B) C) D) Show/Hide Answer A) Incorrect. This expression is not simplified yet. Recall that –a can also be written –a1. Multiply –a1 by a8 to arrive at the correct answer. The correct answer is . B) Incorrect. Do not add the exponents of 2 and 4 together. The Power Rule states that for a power of a power you multiply the exponents. The correct answer is . C) Incorrect. Do not add the exponents of 2 and 4 together. The Power Rule states that for a power of a power you multiply the exponents. The correct answer is . D) Correct. Using the Power Rule, .

The Quotient Rule for Exponents

Let’s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.

You can rewrite the expression as: . Then you can cancel the common factors of 4 in the numerator and denominator: Finally, this expression can be rewritten as 43 using exponential notation. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2.

1. So,  = 45-2 = 43.
2. Be careful that you subtract the exponent in the denominator from the exponent in the numerator.
3. 4. or
5.   = x7−9 = x-2
6. So, to divide two exponential terms with the same base, subtract the exponents.

Notice that  = 40. And we know that   =  = 1. So this may help to explain why 40 = 1.

 Example Problem Evaluate. These two exponents have the same base, 4. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator. = 45

When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.

 Example Problem Simplify. Separate into numerical and variable factors. Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables. Answer =

All of these rules of exponents—the Product Rule, the Power Rule, and the Quotient Rule—are helpful when evaluating expressions with common bases.

 Example Problem Evaluate  when x = 4. Separate into numerical and variable factors. Divide coefficients, and subtract the exponents of the variables. Simplify. Substitute the value 4 for the variable x. Answer = 768

Usually, it is easier to simplify the expression before substituting any values for your variables, but you will get the same answer either way.

 Example Problem Simplify. Use the order of operations with PEMDAS: E: Evaluate exponents. Use the Power Rule to simplify (a5)3. M: Multiply, using the Product Rule as the bases are the same. D: Divide using the Quotient Rule. =

There are rules that help when multiplying and dividing exponential expressions with the same base. To multiply two exponential terms with the same base, add their exponents. To raise a power to a power, multiply the exponents. To divide two exponential terms with the same base, subtract the exponents.

## Algebra Basics – Exponents – In Depth

Exponents are used in many algebra problems, so it's important that you understand the rules for working with exponents. Let's go over each rule in detail, and see some examples.

Rules of 1

There are two simple “rules of 1” to remember.

First, any number raised to the power of “one” equals itself. This makes sense, because the power shows how many times the base is multiplied by itself. If it's only multiplied one time, then it's logical that it equals itself.

Secondly, one raised to any power is one. This, too, is logical, because one times one times one, as many times as you multiply it, is always equal to one.

Product Rule

The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut! Power Rule

The “power rule” tells us that to raise a power to a power, just multiply the exponents. Here you see that 52 raised to the 3rd power is equal to 56. Quotient Rule

The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. You can see why this works if you study the example shown. • Zero Rule
• According to the “zero rule,” any nonzero number raised to the power of zero equals 1.
• Negative Exponents
• The last rule in this lesson tells us that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power.

## Rules for Exponents

• Product and Quotient Rules
• Use the product rule to multiply exponential expressions
• Use the quotient rule to divide exponential expressions
• The Power Rule for Exponents
• Use the power rule to simplify expressions involving products, quotients, and exponents
• Negative and Zero Exponents
• Define and use the zero exponent rule
• Define and use the negative exponent rule
• Simplify Expressions Using the Exponent Rules
• Simplify expressions using a combination of the exponent rules
• Simplify compound exponential expressions with negative exponents Repeated Image

We use exponential notation to write repeated multiplication. For example [latex]10cdot10cdot10[/latex] can be written more succinctly as [latex]10^{3}[/latex]. The 10 in [latex]10^{3}[/latex] is called the base.

The 3 in [latex]10^{3}[/latex] is called the exponent. The expression [latex]10^{3}[/latex] is called the exponential expression.

Knowing the names for the parts of an exponential expression or term will help you learn how to perform mathematical operations on them.

[latex] ext{base}
ightarrow10^{3leftarrow ext{exponent}}[/latex]

[latex]10^{3}[/latex] is read as “10 to the third power” or “10 cubed.” It means [latex]10cdot10cdot10[/latex], or 1,000.

[latex]8^{2}[/latex] is read as “8 to the second power” or “8 squared.” It means [latex]8cdot8[/latex], or 64.

[latex]5^{4}[/latex] is read as “5 to the fourth power.” It means [latex]5cdot5cdot5cdot5[/latex], or 625.

[latex]b^{5}[/latex] is read as “b to the fifth power.” It means [latex]{b}cdot{b}cdot{b}cdot{b}cdot{b}[/latex]. Its value will depend on the value of b.

The exponent applies only to the number that it is next to. Therefore, in the expression [latex]xy^{4}[/latex], only the y is affected by the 4. [latex]xy^{4}[/latex] means [latex]{x}cdot{y}cdot{y}cdot{y}cdot{y}[/latex]. The x in this term is a coefficient of y.

• If the exponential expression is negative, such as [latex]−3^{4}[/latex], it means [latex]–left(3cdot3cdot3cdot3
ight)[/latex] or [latex]−81[/latex].
• If [latex]−3[/latex] is to be the base, it must be written as [latex]left(−3
ight)^{4}[/latex], which means [latex]−3cdot−3cdot−3cdot−3[/latex], or 81.

## Multiplication with Exponents

SfC Home > Arithmetic > Algebra >

by Ron Kurtus (revised 8 July 2019)

When you multiply exponential expressions, there are some simple rules to follow. If they have the same base, you simply add the exponents.

Note: The base of the exponential expression xy is x and the exponent is y.

This is also true for numbers and variables with different bases but with the same exponent. You can apply the rules when other numbers are included.

This rule does not apply when the numbers or variables have different bases and different exponents.

Questions you may have include:

• How do you multiply exponents with the same base?
• What about different bases but with same exponent?
• What about with other numbers?
• When does the rule not apply?

This lesson will answer those questions.

When you multiply two variables or numbers that have the same base, you simply add the exponents.

(xa)*(xb) = xa+b

Thus x3*x4 = x3+4 = x7.

Proof: Since x3 = x*x*x and x4 = x*x*x*x, then

(x*x*x)*(x*x*x*x) = x*x*x*x*x*x*x = x7

### Demonstration with numbers

A demonstration of that rule is seen when you multiply 73 times 72. The result is:

(7*7*7)*(7*7) =

7*7*7*7*7 = 75

Instead of writing out the numbers, you can simply add the exponents:

73*72= 73+2 = 75

## How to Multiply Exponents

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1. 1

Make sure the exponents have the same base. The base is the large number in the exponential expression. You can only use this method if the expressions you are multiplying have the same base.

• For example, you can use this method to multiply 52×53{displaystyle 5^{2} imes 5^{3}}, because they both have the same base (5). On the other hand, you cannot use this method to multiply 52×23{displaystyle 5^{2} imes 2^{3}}, because they have different bases (5 and 2).
2. 2

Add the exponents together. Rewrite the expression, keeping the same base but putting the sum of the original exponents as the new exponent.

• For example, if you are multiplying 52×53{displaystyle 5^{2} imes 5^{3}}, you would keep the base of 5, and add the exponents together:52×53{displaystyle 5^{2} imes 5^{3}}=52+3{displaystyle =5^{2+3}}=55{displaystyle =5^{5}}
3. 3

Calculate the expression. An exponent tells you how many times to multiply a number by itself. You can use a calculator to easily calculate an exponential expression, but you can also calculate by hand.

• For example 55=5×5×5×5×5{displaystyle 5^{5}=5 imes 5 imes 5 imes 5 imes 5}55=3,125{displaystyle 5^{5}=3,125}So, 52×53=3,125{displaystyle 5^{2} imes 5^{3}=3,125}
1. 1

Calculate the first exponential expression. Since the exponents have different bases, there is no shortcut for multiplying them. Calculate the exponent using a calculator or by hand. Remember, an exponent tells you how many times to multiply a number by itself.

• For example, if you are multiplying 23×45{displaystyle 2^{3} imes 4^{5}}, you should note that they do not have the same base. So, you will first calculate 23=2×2×2=8{displaystyle 2^{3}=2 imes 2 imes 2=8}.
2. 2

Calculate the second exponential expression. Do this by multiplying the base number by itself however many times the exponent says.

• For example, 45=4×4×4×4×4=1024{displaystyle 4^{5}=4 imes 4 imes 4 imes 4 imes 4=1024}
3. 3

Rewrite the problem using the new calculations. Following the same example, your new problem becomes 8×1024{displaystyle 8 imes 1024}.

4. 4

Multiply the two numbers. This will give you the final answer to the problem.

• For example: 8×1024=8192.{displaystyle 8 imes 1024=8192.} So, 23×45=8,192{displaystyle 2^{3} imes 4^{5}=8,192}.
1. 1

Multiply the coefficients. Multiply these as you would any whole numbers. Move the number to the outside of the parentheses.

• For example, if multiplying (2x3y5)(8xy4){displaystyle (2x^{3}y^{5})(8xy^{4})}, you would first calculate ((2)x3y5)((8)xy4)=16(x3y5)(xy4){displaystyle ((2)x^{3}y^{5})((8)xy^{4})=16(x^{3}y^{5})(xy^{4})}.
2. 2

Add the exponents of the first variable. Make sure you are only adding the exponents of terms with the same base (variable). Don’t forget that if a variable shows no exponent, it is understood to have an exponent of 1.

• For example:16(x3y5)(xy4)=16(x3)y5(x)y4=16(x3+1)y5y4=16(x4)y5y4{displaystyle 16(x^{3}y^{5})(xy^{4})=16(x^{3})y^{5}(x)y^{4}=16(x^{3+1})y^{5}y^{4}=16(x^{4})y^{5}y^{4}}
3. 3

Add the exponents of the remaining variables. Take care to add exponents with the same base, and don’t forget that variables with no exponents have an understood exponent of 1.

• For example: 16(x4)y5y4=16x4y5+4=16x4y9{displaystyle 16(x^{4})y^{5}y^{4}=16x^{4}y^{5+4}=16x^{4}y^{9}}

• Question What is the solution for 3.5 x 10 to the fourth power? 10^4 = 10 x 10 x 10 x 10 = 10,000, so you are really multiplying 3.5 x 10,000. The shortcut is that, when 10 is raised to a certain power, the exponent tells you how many zeros. 10^4 = 1 followed by 4 zeros = 10,000. Thus, you can just move the decimal point to the right 4 spaces: 3.5 x 10^4 = 35,000.
• Question How do I divide exponents that don't have the same base? To learn how to divide exponents, you can read the following article: http://www.wikihow.com/Divide-Exponents
• Question How do I write 0.0321 in scientific notation? 0.0321 = 3.21 x 10^(-2).
• Question How do I multiply 6.56 x 10^-3? Just move the decimal point three places to the left. 6.56 x 10^(-3) = 0.00656.
• Question How can I calculate the value of \$1000 with annual interest of 9% over 40 years? It depends on how often the interest is compounded. Assuming an annual compounding, the formula is (1,000)(1.09)^40. So you would have to raise 1.09 to the 40th power, then multiply by \$1,000. If the compounding is monthly, the formula is (1,000)(1.0075)^480. So you would raise 1.0075 to the power of 480 before multiplying by \$1,000. Obviously you'd need a calculator for this.
• Question How would I solve (r^3)(3^3)? I'm confused even after reading the article. (r³)(3³) = (3³)(r³) = 3³r³ = 27r³.
• Question What is g^6 .g^3.g^2=? Add exponents: g^11.

• Thanks!

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