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Exponential Terms Raised to an Exponent

Multiply to raise exponents to other exponents

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Power Properties of Exponents

There are 1,000 bacteria present in a culture. When the culture is treated with an antibiotic, the bacteria count is halved every 4 hours. How many bacteria remained 24 hours later?

Power Properties of Exponents

The last set of properties to explore are the power properties. Let’s investigate what happens when a power is raised to another power.

Power of a Power Property

Step 1: Rewrite as five times.

Step 2: Expand each . How many 2’s are there?

Step 3: What is the product of the powers?

Step 4: Fill in the blank.

The other two exponent properties are a form of the distributive property.

Power of a Product Property:

Power of a Quotient Property:




Let's simplify the following.

  1. (do not leave any negative exponents)

This problem uses the Negative Exponent Property. Distribute the power first and then move the negative power of from the numerator to the denominator.

  1.  (do not leave any negative exponents)

This problem is definitely as complicated as these types of problems get. Here, all the properties of exponents will be used. Remember that dividing by a fraction is the same as multiplying by its reciprocal.


Example 1

Earlier, you were asked to find the number of bacteria that remained 24 hours later. 

To find the number of bacteria remaining, we use the exponential expression where n is the number of four-hour periods.

There are 6 four-hour periods in 24 hours, so we set n equal to 6 and solve.

Applying the Power of a Quotient Property, we get:

Therefore, there are 15.625 bacteria remaining after 24 hours.

Example 2

Simplify without negative exponents.

Distribute the 7 to every power within the parenthesis.

Example 3

Simplify without negative exponents: .

Distribute the -3 and 2 to their respective parenthesis and then use the properties of negative exponents, quotient and product properties to simplify.

Example 4

Simplify without negative exponents: .

Distribute the exponents that are outside the parenthesis and use the other properties of exponents to simplify. Anytime a fraction is raised to the -1 power, it is equal to the reciprocal of that fraction to the first power.


Simplify the following expressions without negative exponents.

  1. Challenge 
  2. Rewrite as a power of 2.
  3. Rewrite as a power of 3.
  4. Solve the equation for .
  5. Solve the equation for .

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.3. 

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Power of a Power Property

The power of a power property states that (a^m)^n = a^{mn}.

Power of a Quotient Property

The power of a quotient property states that \left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}.

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