Find the value of the expression.
$\begin{aligned}
x^2+20 y^3 & =9^2+20(2)^3 \\
& =\quad ?
\end{aligned}$
We are tasked to evaluate the mathematical expression \(x^2 + 20y^3 = 9^2 + 20(2)^3\). To solve this, we must follow the order of operations, often referred to by the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). First, we evaluate the exponent terms. The expression consists of a squared term \(9^2\) and a cubed term \(2^3\). Once these powers are evaluated, we perform the necessary multiplication and finally the addition to find the total value.
First, we calculate the exponentiated values:
Now, substitute these back into the expression:
\(81 + 20(8)\).
Following the order of operations, we next perform the multiplication:
\(20 \times 8 = 160\).
Now, the expression simplifies to the sum of the previously calculated values:
\(81 + 160\).
Finally, we perform the addition to find the result:
\(81 + 160 = 241\).
Thus, the value of the expression is \(\boxed{241}\).
This type of multi-step arithmetic is fundamental in science and engineering. For example, in physics, the kinetic energy of an object is proportional to the square of its velocity, \(E = \frac{1}{2}mv^2\). Similarly, when modeling population growth or financial compound interest, terms are often raised to a power (e.g., \(P(1+r)^n\)). Mastering the order of operations ensures that such calculations are performed accurately, preventing errors that could lead to incorrect predictions or financial loss.