Solve the equation with rational exponents. Check all proposed solutions.
[tex]8 x^{\frac{3}{2}}-88=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 (2)
Isolate the term with the rational exponent: 8 x 2 3 = 88 .
Divide both sides by 8: x 2 3 = 11 .
Raise both sides to the power of 3 2 : x = ( 11 ) 3 2 .
The solution set is { 1 1 3 2 } .
Explanation
Problem Analysis We are given the equation 8 x 2 3 − 88 = 0 and asked to solve for x .
Isolating the Exponential Term First, we isolate the term with the rational exponent by adding 88 to both sides of the equation: 8 x 2 3 = 88
Simplifying the Equation Next, we divide both sides by 8 to further isolate the exponential term: x 2 3 = 8 88 = 11
Solving for x To solve for x , we raise both sides of the equation to the power of 3 2 : x = ( 11 ) 3 2
Simplified Solution We can rewrite this as: x = 1 1 3 2
Checking the Solution Now, we check our solution by substituting it back into the original equation: 8 ( 1 1 3 2 ) 2 3 − 88 = 8 ( 11 ) − 88 = 88 − 88 = 0
The solution is valid.
Final Answer Therefore, the solution set is { 1 1 3 2 } .
Examples
Rational exponents are used in various fields, such as physics and engineering, to model relationships between quantities. For example, the period of a simple pendulum is proportional to the square root of its length, which can be expressed using a rational exponent. Solving equations with rational exponents allows us to determine unknown quantities in these models, such as the length of a pendulum given its period. Understanding rational exponents is also crucial in financial mathematics, where they are used to calculate compound interest and other financial metrics. For instance, if you invest money at a certain interest rate compounded quarterly, the growth of your investment can be modeled using rational exponents.
The solution set to the equation 8 x 2 3 − 88 = 0 is { 1 1 3 2 } . This was found by isolating the exponent, simplifying, and verifying the result. Thus, option A is correct.
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