Evaluating the Mathematical Expression

The question asks us to find the value of the expression $2^{2}+ \sqrt{ 10^{2} } -9 \sqrt{16} -17$. To solve this, we need to evaluate each part of the expression following the order of operations (PEMDAS/BODMAS).

Step 1: Calculate Exponents

First, we evaluate the terms with exponents:

• $2^{2}$ means 2 multiplied by itself: $2 \times 2 = 4$.
• $10^{2}$ means 10 multiplied by itself: $10 \times 10 = 100$.

So, the expression now looks like: $4 + \sqrt{ 100 } -9 \sqrt{16} -17$.

Step 2: Calculate Square Roots

Next, we find the square root of the numbers under the radical sign:

• $\sqrt{ 100 }$ is the number which, when multiplied by itself, equals 100. This is 10.
• $\sqrt{ 16 }$ is the number which, when multiplied by itself, equals 16. This is 4.

Substituting these values back into the expression, we get: $4 + 10 -9 \times 4 -17$.

Step 3: Perform Multiplication

Now, we perform the multiplication:

• $9 \times 4 = 36$.

The expression simplifies to: $4 + 10 - 36 - 17$.

Step 4: Perform Addition and Subtraction

Finally, we perform the addition and subtraction from left to right:

• First, add 4 and 10: $4 + 10 = 14$.
• Now the expression is: $14 - 36 - 17$.
• Subtract 36 from 14: $14 - 36 = -22$.
• Now the expression is: $-22 - 17$.
• Subtract 17 from -22: $-22 - 17 = -39$.

Final Result

The value of the expression $2^{2}+ \sqrt{ 10^{2} } -9 \sqrt{16} -17$ is $-39$.