USING TOOLS Let $$ f(x)=7 x^{5 / 3} $$ and $$ g(x)=49 x^{2 / 3} $$. Use a graphing calculator to evaluate $$ (f+g)(x),(f-g)(x),(f g)(x) $$, and $$ \left(\frac{f}{g}\right)(x) $$ when $$ x=5 $$. Round your answers to two decimal places. $$ (f+g)(5)=\square $$ $$ (f-g)(5)=\square $$ $$ (f g)(5)=\square $$ $$ \left(\frac{f}{g}\right)(5)=\square $$

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Let's evaluate the functions $$ f(x) = 7x^{5/3} $$ and $$ g(x) = 49x^{2/3} $$ at $$ x = 5 $$ and compute the required expressions.

### Step 1: Calculate $$ x^{1/3} $$, $$ x^{2/3} $$, and $$ x^{5/3} $$ at $$ x = 5 $$

1. **Cube Root of 5:**
   $$
   5^{1/3} \approx 1.709975947
   $$
2. **$$ 5^{2/3} $$:**
   $$
   5^{2/3} = (5^{1/3})^2 \approx (1.709975947)^2 \approx 2.924017738
   $$
3. **$$ 5^{5/3} $$:**
   $$
   5^{5/3} = 5 \times 5^{2/3} \approx 5 \times 2.924017738 \approx 14.62008869
   $$

### Step 2: Evaluate $$ f(5) $$ and $$ g(5) $$

1. **$$ f(5) $$:**
   $$
   f(5) = 7 \times 14.62008869 \approx 102.34
   $$
2. **$$ g(5) $$:**
   $$
   g(5) = 49 \times 2.924017738 \approx 143.28
   $$

### Step 3: Compute the Required Expressions

1. **Sum $$ (f + g)(5) $$:**
   $$
   (f + g)(5) = f(5) + g(5) \approx 102.34 + 143.28 = 245.62
   $$

2. **Difference $$ (f - g)(5) $$:**
   $$
   (f - g)(5) = f(5) - g(5) \approx 102.34 - 143.28 = -40.94
   $$

3. **Product $$ (f \cdot g)(5) $$:**
   $$
   (f \cdot g)(5) = f(5) \times g(5) \approx 102.34 \times 143.28 \approx 14,663.04
   $$

4. **Quotient $$ \left(\frac{f}{g}\right)(5) $$:**
   $$
   \left(\frac{f}{g}\right)(5) = \frac{f(5)}{g(5)} \approx \frac{102.34}{143.28} \approx 0.71
   $$

### Final Answers

$$
\begin{align*}
(f + g)(5) &= 245.62 \
(f - g)(5) &= -40.94 \
(f \cdot g)(5) &= 14,\!663.04 \
\left(\frac{f}{g}\right)(5) &= 0.71
\end{align*}
$$
$$ \begin{align*} (f + g)(5) &= 245.62 \ (f - g)(5) &= -40.94 \ (f \cdot g)(5) &= 14,\!663.04 \ \left(\frac{f}{g}\right)(5) &= 0.71 \end{align*} $$

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