How to multiply numbers with exponents

  •  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.

How to Multiply Numbers with Exponents

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.)

How to Multiply Numbers with Exponents

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

How to Multiply Numbers with Exponents

And we can conclude that: 3^3 x 3^2 = 3^5

But Why?

How to Multiply Numbers with Exponents

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:

How to Multiply Numbers with Exponents

Here are a few examples applying the multiplying exponents rule:

How to Multiply Exponents

You can multiply many exponential expressions together without having to change their form into the big or small numbers they represent. When multiplying exponents, the only requirement is that the bases of the exponential expressions have to be the same. So, you can multiply

How to Multiply Numbers with Exponents

because the bases are not the same (although the exponents are).

To multiply powers of the same base, add the exponents together:

How to Multiply Numbers with Exponents

If there’s more than one base in an expression with powers, you can combine the numbers with the same bases, find the values, and then write them all together. For example,

How to Multiply Numbers with Exponents

Here’s an example with a number that has no exponent showing:

How to Multiply Numbers with Exponents

When there’s no exponent showing, such as with y, you assume that the exponent is 1, so in the above example, you write

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!

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How to Multiply Numbers with Exponents

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.

How to Multiply Numbers with Exponents

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.

How to Multiply Numbers with Exponents

  • 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.
  • back to top

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

How to Multiply Numbers with 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.

How to Multiply Exponents

  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.[1]

    • 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.[2] 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.[3]

    • 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}}
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Add New Question

  • 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:
  • 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.

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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

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