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Exponents with Fractional Bases

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  • 0:01 Fractions With Exponents
  • 1:10 How to Evaluate
  • 2:13 Example 1
  • 3:07 Example 2
  • 4:07 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can evaluate fractions that have exponents attached to them. Learn how the numerator and denominator behave with exponents.

Fractions with Exponents

In this video lesson, we talk about fractions with exponents. Fractions are the numbers made up of an integer divided by another integer. Exponents are the number that a certain number is raised to. (1/2)^3, (3/4)^10, and (2/9)^4 are all examples of fractions with exponents. In math, we can also say that these are exponents with fractional bases. The base is the number that is being raised to our power.

For the fraction with an exponent (1/2)^3, the 1/2 is the base and the 3 is the exponent. Believe it or not, these problems are used in the real world to calculate things that happen in the real world. Scientists use these problems to calculate the decay of certain items. For example, we can say that light decays as you get further away from the light source. The further you go away from the light, the darker it gets. This decay of the light can be calculated mathematically with an exponent with a fractional base.

How to Evaluate

So, now that we know how these problems are used in the real world, let's see how scientists evaluate them. The process is straightforward and easy. The only rule that you have to remember is that when you see a fraction raised to a power, this power is applied to both the numerator and denominator.

So, you can actually rewrite each problem so that the numerator is raised to the power and the denominator is raised to the power as well. And then you evaluate the numerator and denominator separately. Then you simplify as much as you can to get your answer.

For example, to evaluate (1/2)^3, you can rewrite this problem as (1^3) / (2^3). See how we applied the power to both the numerator and denominator? Evaluating the numerator and denominator separately, you get 1/8 for your answer. This is as simplified as you can get, so this is your final answer.

Let's look at a couple more examples.

Example 1

Evaluate (3/4)^10

Looking at this problem, we see that our fractional base is 3/4 and our exponent is 10. So our fraction 3/4 is being raised to the power of 10. Rewriting this and applying the power to both the numerator and denominator, we get (3^10) / (4^10). Evaluating the numerator and denominator, we get 59,049/1,048,576. Can we simplify this? No, so this means that we have found our answer. (3/4)^10 evaluates to 59,049/1,048,576.

Let's look at another one.

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