Find the value of
`root(3)(175.616)+root(3)(0.175616)+root(3)(0.000175616)`

Options

A. 0.168
B. 62.16
C. 6.216
D. 6.116

Answer

To solve the expression \( \sqrt[3]{175.616} + \sqrt[3]{0.175616} + \sqrt[3]{0.000175616} \), we can follow these steps:

### Step 1: Rewrite the expression using exponent notation
We can express the cube roots in terms of exponents:
\[ \sqrt[3]{175.616} = 175.616^{1/3}, \quad \sqrt[3]{0.175616} = 0.175616^{1/3}, \quad \sqrt[3]{0.000175616} = 0.000175616^{1/3} \]

### Step 2: Express decimal numbers in fraction form
Next, we can rewrite the decimal numbers as fractions:
\[ 175.616 = \frac{175616}{1000}, \quad 0.175616 = \frac{175616}{1000000}, \quad 0.000175616 = \frac{175616}{1000000000} \]

### Step 3: Rewrite the cube roots with the fractions
Now we can rewrite the cube roots:
\[ \sqrt[3]{175.616} = \left(\frac{175616}{1000}\right)^{1/3}, \quad \sqrt[3]{0.175616} = \left(\frac{175616}{1000000}\right)^{1/3}, \quad \sqrt[3]{0.000175616} = \left(\frac{175616}{1000000000}\right)^{1/3} \]

### Step 4: Simplify each term
Using the property of exponents, we can simplify each term:
\[ \sqrt[3]{175.616} = \frac{175616^{1/3}}{10^{1}} = \frac{175616^{1/3}}{10}, \quad \sqrt[3]{0.175616} = \frac{175616^{1/3}}{10^{2}} = \frac{175616^{1/3}}{100}, \quad \sqrt[3]{0.000175616} = \frac{175616^{1/3}}{10^{3}} = \frac{175616^{1/3}}{1000} \]

### Step 5: Combine the terms
Now we can combine the terms:
\[ \sqrt[3]{175.616} + \sqrt[3]{0.175616} + \sqrt[3]{0.000175616} = \frac{175616^{1/3}}{10} + \frac{175616^{1/3}}{100} + \frac{175616^{1/3}}{1000} \]

### Step 6: Factor out \( 175616^{1/3} \)
Factoring out \( 175616^{1/3} \):
\[ = 175616^{1/3} \left( \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} \right) \]

### Step 7: Calculate the sum inside the parentheses
Calculating the sum:
\[ \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} = \frac{100}{1000} + \frac{10}{1000} + \frac{1}{1000} = \frac{111}{1000} \]

### Step 8: Final expression
Thus, we have:
\[ = 175616^{1/3} \cdot \frac{111}{1000} \]

### Step 9: Calculate \( 175616^{1/3} \)
Now we need to calculate \( 175616^{1/3} \). The cube root of \( 175616 \) is \( 56 \) (since \( 56^3 = 175616 \)):
\[ = 56 \cdot \frac{111}{1000} = \frac{6216}{1000} = 6.216 \]

### Conclusion
The value of the expression \( \sqrt[3]{175.616} + \sqrt[3]{0.175616} + \sqrt[3]{0.000175616} \) is \( 6.216 \).