If the mean of three numbers a, b and c is 3, then
`root(3)((7^(a+b-c))(7^(b+c-a))(7^(c+a-b))` equals

Options

A. `7^(1/3)`
B. `7^(2/3)`
C. `7^2`
D. `7^3`

Answer

To solve the expression given in the question, we start with the information provided:

1. **Mean of a, b, and c**: We know that the mean of three numbers \( a, b, c \) is given as 3. This can be expressed mathematically as:
\[ \frac{a + b + c}{3} = 3 \]
Multiplying both sides by 3 gives:
\[ a + b + c = 9 \]

2. **Expression to evaluate**: We need to evaluate the expression:
\[ \sqrt[3]{7^{(a+b-c)} \cdot 7^{(b+c-a)} \cdot 7^{(c+a-b)}} \]

3. **Combine the powers of 7**: Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we can combine the terms inside the cube root:
\[ \sqrt[3]{7^{(a+b-c) + (b+c-a) + (c+a-b)}} \]

4. **Simplifying the exponent**: Now we simplify the exponent:
\[ (a+b-c) + (b+c-a) + (c+a-b) \]
When we expand this, we have:
\[ = a + b - c + b + c - a + c + a - b \]
Notice that \( -c + c = 0 \), \( -a + a = 0 \), and \( -b + b = 0 \). Thus, all terms cancel out, leaving us with:
\[ = a + b + c \]

5. **Substituting the value of \( a + b + c \)**: From step 1, we know that \( a + b + c = 9 \). Therefore, we substitute this value into our expression:
\[ \sqrt[3]{7^{9}} \]

6. **Evaluating the cube root**: The cube root of \( 7^9 \) can be simplified as follows:
\[ \sqrt[3]{7^{9}} = 7^{9/3} = 7^{3} \]

7. **Final result**: Thus, the final result is:
\[ 7^3 = 343 \]

### Summary of the Solution Steps:
1. Calculate \( a + b + c \) from the mean.
2. Write the expression using the property of exponents.
3. Simplify the exponent by combining like terms.
4. Substitute the value of \( a + b + c \).
5. Simplify the cube root.
6. State the final result.