Problem 51 - Algebra 1

Question

Evaluate the expression. \(3 y^{2}+w\) when \(y=8\) and \(w=27\)

Step-by-Step Solution

Answer

The final result of the given expression \(3 y^{2}+w\) for \(y=8\) and \(w=27\) is 219

Step 1: Substitute y

Replace 'y' in the expression with its given value, \(8\). The expression then becomes \(3 * 8^{2} + w\)

Step 2: Calculate Power

Now calculate the power of '8' that is \(8^{2}\). This results in 64. The expression now simplifies to \(3 * 64 + w\)

Step 3: Multiply by 3

Next, multiply '64' by '3'. This results in 192. The expression now simplifies to \(192 + w\)

Step 4: Substitute w

Lastly, replace 'w' in the expression with the given value, '27'. The expression now becomes \(192 + 27\)

Step 5: Final Calculation

Adding '192' and '27' gives the final result of '219'

Key Concepts

Substitution

Substitution is all about replacing variables with their given values to simplify expressions. In our exercise, we are provided with the expression \(3y^{2} + w\) and the values \(y = 8\) and \(w = 27\). The first step in solving this is to substitute the given value of \(y\) into the expression.

By substituting \(y\) with \(8\), the expression becomes \(3 \times 8^{2} + w\). Substituting values helps replace abstract symbols (like \(y\) and \(w\)) with concrete numbers. This makes it easier to perform further operations. Later, we'll also substitute \(w\) with \(27\) to complete our equation. Substitution is essential when solving algebraic expressions as it prepares the equation for calculation.

• Identify the variable given in the expression
• Insert its numerical value into the equation
• Perform the operations step-by-step

Power Calculations

After substitution, we need to handle power calculations. In mathematical terms, 'raising to a power' means multiplying a number by itself. In our exercise, the term \(8^2\) represents \(8\) multiplied by \(8\).

This step is crucial because it will drastically change the value of the expression. Calculating \(8^2\) gives us \(64\). Now the expression looks like \(3 \times 64 + w\). Understanding power calculations involves recognizing the exponent as the number of times the base is multiplied by itself. It's an important skill in algebra.

• Identify the base number and the exponent
• Multiply the base number by itself, exponent times
• Use the result in the next operations

Arithmetic Operations

Once we have performed substitution and calculated the powers, the next step involves arithmetic operations. These are the addition, subtraction, multiplication, and division steps that further break down our expression.

In this exercise, after calculating \(8^2\) and obtaining \(64\), the problem requires multiplying \(64\) by \(3\). This results in \(192\), reducing our expression to \(192 + w\). Arithmetic operations are the basic calculations that help simplify expressions. Next, we substitute \(w\) with \(27\), leading us to the final arithmetic operation: adding \(192\) and \(27\), to get \(219\).

• Execute multiplication and division first
• Proceed with addition and subtraction
• Simplify the expression to reach the final result