Solve [tex]x^{\frac{4}{3}}=16[/tex]
Answer (2)
To solve x 3 4 = 16 , we can raise both sides to the power of 4 3 to isolate x , which gives us x = 1 6 4 3 . Simplifying this leads to the conclusion that x = 8 . Therefore, the final answer is x = 8 .
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Raise both sides of the equation to the power of 4 3 to isolate x : x = 1 6 4 3 .
Rewrite 16 as 2 4 : x = ( 2 4 ) 4 3 .
Simplify the exponent: x = 2 4 ⋅ 4 3 = 2 3 .
Calculate 2 3 to find the value of x : x = 8 . Therefore, the solution is 8 .
Explanation
Understanding the Problem We are given the equation x 3 4 = 16 and asked to solve for x .
Isolating x To isolate x , we raise both sides of the equation to the power of 4 3 . This gives us ( x 3 4 ) 4 3 = 1 6 4 3 x = 1 6 4 3
Rewriting 16 We can rewrite 16 as 2 4 , so we have x = ( 2 4 ) 4 3
Simplifying the Exponent Now we simplify the exponent: x = 2 4 ⋅ 4 3 = 2 3
Calculating the Value of x Finally, we calculate 2 3 :
x = 2 3 = 8
Final Answer Therefore, the solution to the equation x 3 4 = 16 is x = 8 .
Examples
Imagine you are designing a spherical tank, and you know that the volume of the tank must be 16 cubic meters. The radius of the sphere is related to the volume by the formula V = 3 4 π r 3 . If you knew the volume was proportional to r 3 4 instead, solving an equation like the one above would help you determine the required radius to achieve the desired volume. This type of problem appears in various engineering and physics applications where relationships between quantities involve fractional exponents.
