%5E2%2F3%3D4.jpeg "(x+5)^2/3=4")
Solving the Equation: (x+5)^(2/3) = 4
This equation involves a fractional exponent, which can be a bit tricky to work with. Let's break it down step-by-step to find the solution.
Understanding Fractional Exponents
A fractional exponent like (2/3) represents both a root and a power. The denominator (3) indicates the root, and the numerator (2) indicates the power. So, (x+5)^(2/3) is equivalent to the cube root of (x+5) squared: ∛[(x+5)²].
Solving for x
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Isolate the term with the fractional exponent: Since the right side of the equation is already a constant, we don't need to perform any additional steps here.
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Raise both sides to the reciprocal power: To get rid of the fractional exponent (2/3), we raise both sides of the equation to the power of 3/2 (the reciprocal of 2/3). This gives us: [(x+5)^(2/3)]^(3/2) = 4^(3/2)
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Simplify: The exponents cancel out on the left side, leaving us with (x+5). On the right side, 4^(3/2) is equivalent to the square root of 4 cubed, which is 8. This gives us: x + 5 = 8
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Solve for x: Subtracting 5 from both sides, we get: x = 3
Verifying the Solution
To ensure our solution is correct, we can substitute x = 3 back into the original equation: (3 + 5)^(2/3) = 8^(2/3) = (8^(1/3))^2 = 2^2 = 4
Since the left and right sides of the equation are equal, we've verified that x = 3 is the correct solution.