What is the solution to the equation?
$3(x+9)^{\frac{3}{4}}=24$
We are given the equation \(3(x+9)^{\frac{3}{4}} = 24\). Our goal is to solve for \(x\) and identify which of the provided options (-3, 6, 7, 25) is the correct solution. The equation involves a power of a linear expression, and our first step is to isolate that power term.
First, we divide both sides of the equation by 3 to isolate the term involving the fractional exponent:
\[(x+9)^{\frac{3}{4}} = \frac{24}{3}\]
Simplifying the right side, we get:
\[(x+9)^{\frac{3}{4}} = 8\]
This is our simplified equation.
To solve for \(x+9\), we raise both sides of the equation to the power of \(\frac{4}{3}\), which is the reciprocal of \(\frac{3}{4}\):
\[\left((x+9)^{\frac{3}{4}}\right)^{\frac{4}{3}} = 8^{\frac{4}{3}}\]
Applying the exponent rules, the left side simplifies to \(x+9\). On the right side, we compute \(8^{\frac{4}{3}}\):
\[8^{\frac{4}{3}} = \left(8^{\frac{1}{3}}\right)^4 = 2^4 = 16\]
Thus, we have:
\[x + 9 = 16\]
Finally, we isolate \(x\) by subtracting 9 from both sides of the equation:
\[x = 16 - 9\]
\[x = 7\]
To verify, we substitute \(x = 7\) back into the original equation:
\[3(7+9)^{\frac{3}{4}} = 3(16)^{\frac{3}{4}} = 3 \times (2^4)^{\frac{3}{4}} = 3 \times 2^3 = 3 \times 8 = 24\]
The result confirms that 7 is the correct solution. Looking at the given options, the correct choice is 7.
In engineering and physics, fractional power equations often model relationships where growth or decay occurs according to a non-linear scale. For example, the stress on a material or the scaling of power relative to a physical dimension can be described by such equations. Solving these allows engineers to determine critical thresholds for structural integrity or performance limits in real-world designs.