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Example Question #1 : Evaluate Expressions With Rational Exponents
Simplify the expression:
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.
The final simplified answer becomes:
Example Question #2 : Evaluate Expressions With Rational Exponents
Evaluate the following expression using knowledge of the properties of exponents:
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 #3 : Evaluate Expressions With Rational Exponents
Evaluate the following expression and solve for .
To solve this problem, recall that you can set exponents equal to eachother if they have the same base.
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 #4 : Evaluate Expressions With Rational Exponents
Solve for .
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 #5 : Evaluate Expressions With Rational Exponents
Solve for .
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 #6 : Evaluate Expressions With Rational Exponents
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 #7 : Evaluate Expressions With Rational Exponents
What is the value of ?
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 .