Solve the equation [tex]$125=x^{\frac{3}{2}}$[/tex].
A. [tex]$x=25$[/tex]
B. [tex]$x=5$[/tex]
C. [tex]$x=-5$[/tex] or [tex]$x=25$[/tex]
D. [tex]$x=5$[/tex] or [tex]$x=25$[/tex]
Answer (2)
Rewrite the equation as x 2 3 = 125 .
Raise both sides to the power of 3 2 to get x = 12 5 3 2 .
Simplify 12 5 3 2 as ( 5 3 ) 3 2 = 5 2 = 25 .
The solution is x = 25 .
Explanation
Understanding the Problem We are given the equation 125 = x 2 3 and asked to solve for x . This means we need to find the value(s) of x that satisfy this equation. We will manipulate the equation to isolate x and then check our solution(s).
Isolating x To isolate x , we can raise both sides of the equation to the power of 3 2 . This will undo the exponent of 2 3 on the x term. So we have ( x 2 3 ) 3 2 = 12 5 3 2 Using the property of exponents that ( a b ) c = a b ⋅ c , we get x 2 3 ⋅ 3 2 = 12 5 3 2 x 1 = 12 5 3 2 x = 12 5 3 2
Evaluating the Expression Now we need to evaluate 12 5 3 2 . We can rewrite 125 as 5 3 , so we have x = ( 5 3 ) 3 2 Using the property of exponents again, we get x = 5 3 ⋅ 3 2 x = 5 2 x = 25
Checking the Solution Now we need to check if our solution is valid. We substitute x = 25 back into the original equation: 125 = 2 5 2 3 125 = ( 2 5 2 1 ) 3 125 = ( 5 ) 3 125 = 125 Since the equation holds true, x = 25 is a valid solution.
Final Answer Therefore, the solution to the equation 125 = x 2 3 is x = 25 . Looking at the given options, the correct answer is A.
Examples
Imagine you are designing a spherical water tank, and you know the volume of water it needs to hold. The equation V = 3 4 π r 3 relates the volume V to the radius r . If you need to find the radius r given a specific volume V , you would solve for r by raising both sides to the power of 3 1 . Similarly, the problem we solved involves finding a value when it's raised to a fractional power, which is a common task in various engineering and scientific applications.
The solution to the equation 125 = x 2 3 is x = 25 after isolating x and verifying the solution. Thus, the correct option is A: x = 25 .
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