Precalculus : Evaluate Expressions With Rational Exponents

Example Questions

Example Question #11 : 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 #2 : Evaluate 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 #1 : Evaluate 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 #1 : Evaluate 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 #2 : Evaluate 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 #1 : Evaluate 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 #61 : Exponential And Logarithmic Functions

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 All Precalculus Resources 