# Precalculus : Rational Exponents

## Example Questions

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### Example Question #11 : Simplify Expressions With Rational Exponents

Simplify the function:

Explanation:

When an exponent is raised to the power of another exponent, just multiply the exponents together.

### Example Question #12 : Simplify Expressions With Rational Exponents

Simplify the function:

Explanation:

When an exponent is raised to the power of another exponent, just multiply the exponents together.

### Example Question #13 : Simplify Expressions With Rational Exponents

Simplify the expression:

.

Explanation:

First, you can begin to simplfy the numerator by converting all 3 expressions into base 2.

, which simplifies to

For the denominator, the same method applies. Convert the 25 into base 5, and when simplified becomes simply 5.

### Example Question #14 : Simplify Expressions With Rational Exponents

Evaluate the following expression using knowledge of the properties of exponents:

Explanation:

Let's work through this equation involving exponents one term at a time. The first term we see is , for which we can apply the following property:

So if we plug our values into the formula for the property, we get:

Because . Our next term is , for which we'll need the property:

Using the values for our term, we have:

The third term of the equation is , for which the quickest way to evaluate would be using the following property:

Using the values from our term, this gives us:

The next property we will need to consider for our fourth term is given below:

If we plug in the corresponding values from our term, we get:

Finally, our last term requires knowledge of the following simple property: Any number raised to the power of zero is 1. With this in mind, our last term becomes:

Rewriting the equation with all of the values we've just evaluated, we obtain our final answer:

### Example Question #15 : Simplify Expressions With Rational Exponents

Evaluate the following expression and solve for .

Explanation:

To solve this problem, recall that you can set exponents equal to eachother if they have the same base.

See below:

So, we have

Because both sides of this equation have a base of seven, we can set the exponents equal to eachother and solve for t.

### Example Question #11 : Simplify Expressions With Rational Exponents

Solve for .

Explanation:

We begin by taking the natural log of the equation:

Simplifying the left side of the equation using the rules of logarithms gives:

We group the x terms to get:

We reincorporate the exponents into the logarithms and use the identity property of the natural log to obtain:

We combine the logarithms using the multiplication/sum rule to get:

We then solve for x:

### Example Question #17 : Simplify Expressions With Rational Exponents

Solve for .

Explanation:

We begin by factoring out the term  to get:

This equation gives our first solution:

Then we check for more solutions:

Therefore our solution is

### Example Question #18 : Simplify Expressions With Rational Exponents

Evaluate   when

Explanation:

Remember the denominator of a rational exponent is equivalent to the index of a root.

This should simplify quite nicely.

When  it gives us,

### Example Question #19 : Simplify Expressions With Rational Exponents

What is the value of ?

15

Explanation:

What does an exponent of one-third mean? Consider our expression and raise it to the third power.

Simplifying, we get:

Thus, we are looking for a number that when cubed, we get . Thus, we are discussing the cube root of , or

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