Solve the equation with rational exponents. Check all proposed solutions.
[tex]7 x^{\frac{9}{4}}-21=0[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is { [ ] }.
(Type an exact answer in simplified form. Use a comma to separate answers as needed.)
B. The solution set is [tex]\varnothing[/tex].
Answer (1)
Isolate the term with the rational exponent: 7 x 4 9 = 21 .
Divide both sides by 7: x 4 9 = 3 .
Raise both sides to the power of 9 4 : ( x 4 9 ) 9 4 = 3 9 4 .
Simplify and rewrite in radical form: x = 3 9 4 = 9 81 .
The solution set is { 9 81 } .
Explanation
Understanding the Problem We are given the equation 7 x 4 9 − 21 = 0 and asked to solve for x . We also need to check our solution to make sure it is not extraneous.
Isolating the Exponential Term First, we isolate the term with the rational exponent by adding 21 to both sides of the equation: 7 x 4 9 = 21
Simplifying the Equation Next, we divide both sides by 7: x 4 9 = 3
Raising to the Reciprocal Power To solve for x , we raise both sides to the power of 9 4 : ( x 4 9 ) 9 4 = 3 9 4
Simplifying the Exponent Simplifying the exponents, we get: x = 3 9 4
Rewriting in Radical Form We can rewrite the solution in radical form: x = 9 3 4 = 9 81
Checking the Solution Now, we check the solution in the original equation: 7 ( 9 81 ) 4 9 − 21 = 7 ( 8 1 9 1 ) 4 9 − 21 = 7 ( 8 1 4 1 ) − 21 = 7 ( 3 ) − 21 = 21 − 21 = 0
Final Answer Since the solution checks, the solution set is { 9 81 }. Therefore, the correct choice is A. The solution set is { 9 81 }.
Stating the Solution Therefore, the solution set is { 9 81 } .
Examples
Rational exponents are used in various fields, such as physics and engineering, to model relationships between different quantities. For example, the period of a pendulum is related to its length by a rational exponent. Understanding how to solve equations with rational exponents allows us to analyze and predict the behavior of such systems. In finance, compound interest calculations often involve rational exponents to determine the growth of investments over time. By mastering these concepts, students can apply them to real-world scenarios and make informed decisions.
