Let's solve the equation step by step. We start with the equation:
[tex]\[ 3(x + 9)^{\frac{3}{4}} = 24 \][/tex]
1. Isolate the term with the variable [tex]\(x\)[/tex]:
Divide both sides of the equation by 3:
[tex]\[ (x + 9)^{\frac{3}{4}} = \frac{24}{3} \][/tex]
Simplifying the right side, we get:
[tex]\[ (x + 9)^{\frac{3}{4}} = 8 \][/tex]
2. Eliminate the exponent:
To get rid of the [tex]\(\frac{3}{4}\)[/tex] exponent, raise both sides to the reciprocal power, which is [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[ ( (x + 9)^{\frac{3}{4}} )^{\frac{4}{3}} = 8^{\frac{4}{3}} \][/tex]
Simplifying the left side, the exponents [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex] cancel out, leaving:
[tex]\[ x + 9 = 8^{\frac{4}{3}} \][/tex]
3. Evaluate the right-hand side:
Calculate [tex]\( 8^{\frac{4}{3}} \)[/tex]:
[tex]\[ 8^{\frac{4}{3}} \approx 16 \][/tex] (The exact result is very close to 16)
So, the equation simplifies to:
[tex]\[ x + 9 = 16 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Subtract 9 from both sides:
[tex]\[ x = 16 - 9 \][/tex]
[tex]\[ x \approx 7 \][/tex]
Therefore, the solution to the equation [tex]\( 3(x + 9)^{\frac{3}{4}} = 24 \)[/tex] is:
[tex]\[ \boxed{7} \][/tex]
So the correct answer is C. 7.
