To solve the equation \(\frac{(x+16)^2}{9} = 7\), we start by eliminating the fraction. We can do this by multiplying both sides by 9:
\[ (x+16)^2 = 7 \times 9 \]
Calculating the right side gives:
\[ (x+16)^2 = 63 \]
Now, we will take the square root of both sides. Remember to consider both the positive and negative roots:
\[ x + 16 = \pm \sqrt{63} \]
Next, we can simplify \(\sqrt{63}\):
\[ \sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7} \]
Now we have two equations to solve:
- \(x + 16 = 3\sqrt{7}\)
- \(x + 16 = -3\sqrt{7}\)
For the first equation:
\[ x + 16 = 3\sqrt{7} \]
Subtracting 16 from both sides gives:
\[ x = 3\sqrt{7} - 16 \]
For the second equation:
\[ x + 16 = -3\sqrt{7} \]
Subtracting 16 from both sides gives:
\[ x = -3\sqrt{7} - 16 \]
Next, we will approximate the values by calculating \(\sqrt{7}\):
\[ \sqrt{7} \approx 2.64575 \]
Now we can find \(3\sqrt{7}\):
\[ 3\sqrt{7} \approx 3 \times 2.64575 \approx 7.93725 \]
Substituting this into the first equation:
\[ x \approx 7.93725 - 16 \approx -8.06275 \]
For the second equation:
\[ -3\sqrt{7} \approx -7.93725 \]
So,
\[ x \approx -7.93725 - 16 \approx -23.93725 \]
Rounding to the nearest hundredth, we have:
- \(x \approx -8.06\)
- \(x \approx -23.94\)
Thus, the solutions to the equation are:
\[ x \approx -8.06 \quad \text{and} \quad x \approx -23.94 \]