What is the value of this expression when $c=-4$ and $d=10$?
$\frac{1}{4}\left(c^3+d^2\right)$
We are given the expression \(\frac{1}{4}(c^3 + d^2)\) and asked to evaluate it for the specific values \(c = -4\) and \(d = 10\). To do this, we need to perform the exponentiation of each variable, add the results, and then divide by 4.
First, we evaluate the cubed term \(c^3\). Substituting \(c = -4\):
\[c^3 = (-4)^3 = (-4) \times (-4) \times (-4) = -64.\]
Next, we evaluate the squared term \(d^2\). Substituting \(d = 10\):
\[d^2 = (10)^2 = 10 \times 10 = 100.\]
Now, we substitute these values back into the original expression \(\frac{1}{4}(c^3 + d^2)\):
\[\frac{1}{4}(-64 + 100) = \frac{1}{4}(36).\]
Finally, dividing 36 by 4:
\[\frac{36}{4} = 9.\]
The value of the expression is 9, which matches the second option.
This type of evaluation is fundamental in many real-world applications, such as physics or engineering. For instance, if you are calculating the kinetic energy of an object based on different components, you might have an expression like \(\frac{1}{2}mv^2\). Knowing how to correctly substitute and calculate variables helps you determine the resulting energy, speed, or other physical quantities, ensuring your calculations are accurate for practical designs or scientific predictions.