The product of `((1)/(5)x^(2)-(1)/(6)y^(2)) and (5x^(2)+6y^(2))` is

Options

A. 1
B. `x^(4)+(11)/(60)x^(2)y^(2)+y^(4)`
C. `x^(4)+(11)/(30)x^(2)y^(2)-y^(4)`
D. `x^(4)-(11)/(30)x^(2)y^(2)-y^(4)`

Answer

To find the product of the expressions \(\left(\frac{1}{5}x^2 - \frac{1}{6}y^2\right)\) and \((5x^2 + 6y^2)\), we will follow these steps:

### Step 1: Distribute the first expression into the second expression
We will use the distributive property (also known as the FOIL method for binomials) to multiply the two expressions:

\[ \left(\frac{1}{5}x^2 - \frac{1}{6}y^2\right)(5x^2 + 6y^2) = \frac{1}{5}x^2(5x^2) + \frac{1}{5}x^2(6y^2) - \frac{1}{6}y^2(5x^2) - \frac{1}{6}y^2(6y^2) \]

### Step 2: Calculate each term
Now we will calculate each term separately:

1. \(\frac{1}{5}x^2 \cdot 5x^2 = x^4\)
2. \(\frac{1}{5}x^2 \cdot 6y^2 = \frac{6}{5}x^2y^2\)
3. \(-\frac{1}{6}y^2 \cdot 5x^2 = -\frac{5}{6}x^2y^2\)
4. \(-\frac{1}{6}y^2 \cdot 6y^2 = -y^4\)

So, we have:

\[ x^4 + \frac{6}{5}x^2y^2 - \frac{5}{6}x^2y^2 - y^4 \]

### Step 3: Combine like terms
Next, we need to combine the like terms \(\frac{6}{5}x^2y^2\) and \(-\frac{5}{6}x^2y^2\). To do this, we need a common denominator.

The least common multiple of 5 and 6 is 30. We can rewrite the fractions:

\[ \frac{6}{5}x^2y^2 = \frac{36}{30}x^2y^2 \]
\[ -\frac{5}{6}x^2y^2 = -\frac{25}{30}x^2y^2 \]

Now, we can combine them:

\[ \frac{36}{30}x^2y^2 - \frac{25}{30}x^2y^2 = \frac{11}{30}x^2y^2 \]

### Step 4: Write the final expression
Now we can write the final expression by combining all the terms:

\[ x^4 + \frac{11}{30}x^2y^2 - y^4 \]

### Final Answer
Thus, the product of the given expressions is:

\[ x^4 + \frac{11}{30}x^2y^2 - y^4 \]

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