If `((9)/(10))^(3) -(2/5)^(3)-(1/2)^(3)=(x)/(50)`, find x

Answer

To solve the equation \(\left(\frac{9}{10}\right)^{3} - \left(\frac{2}{5}\right)^{3} - \left(\frac{1}{2}\right)^{3} = \frac{x}{50}\), we will follow these steps:

### Step 1: Identify the values
Let:
- \( x = \frac{9}{10} \)
- \( y = -\frac{2}{5} \)
- \( z = -\frac{1}{2} \)

### Step 2: Check if \( x + y + z = 0 \)
We need to verify if \( x + y + z = 0 \):
\[ x + y + z = \frac{9}{10} - \frac{2}{5} - \frac{1}{2} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 10, 5, and 2 is 10. We can rewrite the fractions:
- \( \frac{9}{10} \) remains the same.
- \( -\frac{2}{5} = -\frac{4}{10} \)
- \( -\frac{1}{2} = -\frac{5}{10} \)

Now substituting these values:
\[ x + y + z = \frac{9}{10} - \frac{4}{10} - \frac{5}{10} = \frac{9 - 4 - 5}{10} = \frac{0}{10} = 0 \]
Thus, \( x + y + z = 0 \) holds true.

### Step 3: Apply the identity
Using the identity for cubes:
\[ x^3 + y^3 + z^3 - 3xyz = 0 \]
We can rearrange it to find:
\[ x^3 + y^3 + z^3 = 3xyz \]

### Step 4: Calculate \( x^3, y^3, z^3 \)
Now we will calculate each cube:
- \( x^3 = \left(\frac{9}{10}\right)^3 = \frac{729}{1000} \)
- \( y^3 = \left(-\frac{2}{5}\right)^3 = -\frac{8}{125} = -\frac{64}{1000} \) (after converting to a common denominator)
- \( z^3 = \left(-\frac{1}{2}\right)^3 = -\frac{1}{8} = -\frac{125}{1000} \) (after converting to a common denominator)

Now substituting these values into the equation:
\[ x^3 + y^3 + z^3 = \frac{729}{1000} - \frac{64}{1000} - \frac{125}{1000} \]
Calculating the sum:
\[ x^3 + y^3 + z^3 = \frac{729 - 64 - 125}{1000} = \frac{540}{1000} = \frac{27}{50} \]

### Step 5: Set equal to \( 3xyz \)
Now we need to calculate \( 3xyz \):
\[ xyz = \left(\frac{9}{10}\right) \left(-\frac{2}{5}\right) \left(-\frac{1}{2}\right) = \frac{9 \cdot 2 \cdot 1}{10 \cdot 5 \cdot 2} = \frac{18}{100} = \frac{9}{50} \]
Thus,
\[ 3xyz = 3 \cdot \frac{9}{50} = \frac{27}{50} \]

### Step 6: Equate and solve for \( x \)
We have:
\[ x^3 + y^3 + z^3 = 3xyz \implies \frac{27}{50} = \frac{x}{50} \]
From this, we can find \( x \):
\[ x = 27 \]

### Final Answer
Thus, the value of \( x \) is \( 27 \).

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