If x = 16, y = 10, z = 5 and t=-1, find the value of `(x-y)(5sqrtx-y)+sqrt((x-y)(z-t))`.

Options

A. 66
B. 72
C. 76
D. 80

Answer

To solve the expression \((x-y)(5\sqrt{x}-y)+\sqrt{(x-y)(z-t)}\) given the values \(x = 16\), \(y = 10\), \(z = 5\), and \(t = -1\), we will follow these steps:

### Step 1: Substitute the values into the expression
We start by substituting the given values into the expression:

\[ (x-y)(5\sqrt{x}-y)+\sqrt{(x-y)(z-t)} = (16-10)(5\sqrt{16}-10)+\sqrt{(16-10)(5-(-1))} \]

### Step 2: Simplify \(x - y\)
Calculate \(x - y\):

\[ 16 - 10 = 6 \]

### Step 3: Calculate \(\sqrt{x}\)
Calculate \(\sqrt{x}\):

\[ \sqrt{16} = 4 \]

### Step 4: Substitute back into the expression
Now we substitute \(x - y\) and \(\sqrt{x}\) back into the expression:

\[ (6)(5 \cdot 4 - 10) + \sqrt{(6)(5 - (-1))} \]

### Step 5: Simplify the first part
Calculate \(5 \cdot 4 - 10\):

\[ 20 - 10 = 10 \]

Now substitute this back into the expression:

\[ (6)(10) + \sqrt{(6)(5 + 1)} \]

### Step 6: Calculate the first part
Now calculate \(6 \cdot 10\):

\[ 60 \]

### Step 7: Simplify the second part
Calculate \(5 + 1\):

\[ 5 + 1 = 6 \]

Now substitute this back into the square root:

\[ \sqrt{(6)(6)} = \sqrt{36} = 6 \]

### Step 8: Combine both parts
Now combine both parts of the expression:

\[ 60 + 6 = 66 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{66} \]