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In algebra, evaluating expressions involves replacing variables with their actual values, then calculating the remaining numerical expression step by step. This process is an essential skill in math, as it allows you to understand and manipulate algebraic formulas. Let's explore how to evaluate an expression with a given example:
Given an algebraic expression like \(3yz^2 + 1\), we are asked to evaluate it for specific values of \(x\), \(y\), and \(z\). In this case, we substitute \(y = 5\) and \(z = 3\) into the expression:

• Replace variable \(y\) with 5.
• Replace variable \(z\) with 3.

After substitution, the expression becomes \(3(5)(3)^2 + 1\).
Now, it's just a matter of performing the arithmetic operations, starting with evaluating the exponent.

Exponents are a way to represent repeated multiplication of the same number. In our example, \(z^2\) signifies \(z\) multiplied by itself. So, if \(z = 3\), then:
\(z^2 = 3^2 = 3 \times 3 = 9\).
Exponents are powerful because they simplify large multiplication operations into smaller, more manageable expressions. Evaluating the exponent is always the next step after substituting the variables.
Once you know the value of the exponent, you can rewrite the substituted expression, transforming \(3(5)(3)^2 + 1\) to \(3 \times 5 \times 9 + 1\), ready for the next arithmetic operation.

The order of operations is a critical foundation in algebra, ensuring everyone evaluates expressions consistently. This order can be memorized with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Let's continue with the expression \(3 \times 5 \times 9 + 1\):
1. **Parentheses:** There are none left to process.2. **Exponents:** Already evaluated.3. **Multiplication/Division:** We handle multiplication first:

• Multiply \(3\) and \(5\) to get \(15\).
• Then multiply \(15\) by \(9\) to yield \(135\).

4. **Addition/Subtraction:** Finally, add 1 to 135:

• So, \(135 + 1 = 136\).

By following these steps, you ensure the expression is evaluated correctly and consistently, arriving at the final result: \(136\). Remember, always follow the order of operations to avoid errors and achieve accurate results.