If `sqrt(x)-sqrt(y)=1`, `sqrt(x)+sqrt(y)=17` then `sqrt(xy)=?`

Options

A. `sqrt(72)`
B. `72`
C. `32`
D. `24`

Answer

To solve the problem, we have the equations:

1. \( \sqrt{x} - \sqrt{y} = 1 \) (Equation 1)
2. \( \sqrt{x} + \sqrt{y} = 17 \) (Equation 2)

We need to find \( \sqrt{xy} \).

### Step 1: Solve for \( \sqrt{x} \) and \( \sqrt{y} \)

Let's denote \( \sqrt{x} = a \) and \( \sqrt{y} = b \). Then we can rewrite the equations as:

1. \( a - b = 1 \) (Equation 1)
2. \( a + b = 17 \) (Equation 2)

### Step 2: Add the two equations

Now, we can add Equation 1 and Equation 2:

\[ (a - b) + (a + b) = 1 + 17 \]

This simplifies to:

\[ 2a = 18 \]

### Step 3: Solve for \( a \)

Dividing both sides by 2 gives:

\[ a = 9 \]

### Step 4: Substitute \( a \) back to find \( b \)

Now, substitute \( a \) back into Equation 1 to find \( b \):

\[ 9 - b = 1 \]

Solving for \( b \):

\[ b = 9 - 1 = 8 \]

### Step 5: Find \( x \) and \( y \)

Now we have \( a = \sqrt{x} = 9 \) and \( b = \sqrt{y} = 8 \).

To find \( x \) and \( y \):

\[ x = a^2 = 9^2 = 81 \]
\[ y = b^2 = 8^2 = 64 \]

### Step 6: Find \( \sqrt{xy} \)

Now we can find \( \sqrt{xy} \):

\[ \sqrt{xy} = \sqrt{81 \times 64} \]

Calculating \( 81 \times 64 \):

\[ 81 \times 64 = 5184 \]

Now, taking the square root:

\[ \sqrt{5184} = 72 \]

### Final Answer

Thus, \( \sqrt{xy} = 72 \).

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