Multiplying exponents with different bases and different powers

How to multiply exponents.

• Multiplying exponents with same base
• Multiplying exponents with different bases
• Multiplying negative exponents
• Multiplying fractions with exponents
• Multiplying fractional exponents
• Multiplying variables with exponents
• Multiplying square roots with exponents

Multiplying exponents with same base

For exponents with the same base, we should add the exponents:

a n ⋅ a m = a n+m

Example:

23 ⋅ 24 = 23+4 = 27 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128

Multiplying exponents with different bases

When the bases are diffenrent and the exponents of a and b are the same, we can multiply a and b first:

a n ⋅ b n = (a ⋅ b) n

Example:

32 ⋅ 42 = (3⋅4)2 = 122 = 12⋅12 = 144

When the bases and the exponents are different we have to calculate each exponent and then multiply:

a n ⋅ b m

Example:

32 ⋅ 43 = 9 ⋅ 64 = 576

Multiplying negative exponents

For exponents with the same base, we can add the exponents:

a -n ⋅ a -m = a -(n+m) = 1 / a n+m

Example:

2-3 ⋅ 2-4 = 2-(3+4) = 2-7 = 1 / 27 = 1 / (2⋅2⋅2⋅2⋅2⋅2⋅2) = 1 / 128 = 0.0078125

When the bases are diffenrent and the exponents of a and b are the same, we can multiply a and b first:

a -n ⋅ b -n = (a ⋅ b) -n

Example:

3-2 ⋅ 4-2 = (3⋅4)-2 = 12-2 = 1 / 122 = 1 / (12⋅12) = 1 / 144 = 0.0069444

When the bases and the exponents are different we have to calculate each exponent and then multiply:

a -n ⋅ b -m

Example:

3-2 ⋅ 4-3 = (1/9) ⋅ (1/64) = 1 / 576 = 0.0017361

Multiplying fractions with exponents

Multiplying fractions with exponents with same fraction base:

(a / b) n ⋅ (a / b) m = (a / b) n+m

Example:

(4/3)3 ⋅ (4/3)2 = (4/3)3+2 = (4/3)5 = 45 / 35 = 4.214

Multiplying fractions with exponents with same exponent:

(a / b) n ⋅ (c / d) n = ((a / b)⋅(c / d)) n

Example:

(4/3)3 ⋅ (3/5)3 = ((4/3)⋅(3/5))3 = (4/5)3 = 0.83 = 0.8⋅0.8⋅0.8 = 0.512

Multiplying fractions with exponents with different bases and exponents:

(a / b) n ⋅ (c / d) m

Example:

(4/3)3 ⋅ (1/2)2 = 2.37 ⋅ 0.25 = 0.5925

Multiplying fractional exponents

Multiplying fractional exponents with same fractional exponent:

a n/m ⋅ b n/m = (a ⋅ b) n/m

Example:

23/2 ⋅ 33/2 = (2⋅3)3/2 = 63/2 = √(63) = √216 = 14.7

Multiplying fractional exponents with same base:

a (n/m) ⋅ a (k/j) = a [(n/m)+(k/j)]

Example:

2(3/2) ⋅ 2(4/3) = 2[(3/2)+(4/3)] = 7.127

Multiplying fractional exponents with different exponents and fractions:

a n/m ⋅ b k/j

Example:

2 3/2 ⋅ 24/3 = √(23) ⋅ 3√(24) = 2.828 ⋅ 2.52 = 7.127

Multiplying square roots with exponents

For exponents with the same base, we can add the exponents:

(√a)n ⋅ (√a)m = a(n+m)/2

Example:

(√5)2 ⋅ (√5)4 = 5(2+4)/2 = 56/2 = 53 = 125

Multiplying variables with exponents

For exponents with the same base, we can add the exponents:

xn ⋅ xm = xn+m

Example:

x2 ⋅ x3 = (x⋅x) ⋅ (x⋅x⋅x) = x2+3 = x5

See also

• Exponents rules
• Dividing exponents
• Adding exponenets
• Exponent calculator

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Exponents are a way to identify numbers that are being multiplied by themselves. They are often called powers. You will come across exponents frequently in algebra, so it is helpful to know how to work with these types of expressions. You can multiply exponential expressions just as you can multiply other numbers. If the exponents have the same base, you can use a shortcut to simplify and calculate; otherwise, multiplying exponential expressions is still a simple operation.

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.

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]

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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.

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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.

2. 

   2

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

   • For example,

3. 

   3

   Rewrite the problem using the new calculations. Following the same example, your new problem becomes .

4. 

   4

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

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1. 

   1

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

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:
     

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:
     

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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: http://www.wikihow.com/Divide-Exponents

• Question

  How do I write 0.0321 in scientific notation?

  

  0.0321 = 3.21 x 10^(-2).

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• Any number or variable with an exponent of 0 is equal to 1. For example, .

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Article SummaryX

If you want to multiply exponents with the same base, simply add the exponents together. For example 7 to the third power × 7 to the fifth power = 7 to the eighth power because 3 + 5 = 8. However, to solve exponents with different bases, you have to calculate the exponents and multiply them as regular numbers. For example, 2 squared = 4, and 3 squared = 9, so 2 squared times 3 squared = 36 because 4 × 9 = 36. To learn how to multiply exponents with mixed variables, read more!

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