Problem 52 - Algebra 1

Question

Evaluate the expression. \(2 s^{3}-3 t\) when \(s=4\) and \(t=6\)

Step-by-Step Solution

Answer

The final result of the expression is \(110\).

Step 1: Substitution of values

To find the answer, substitute the given values into the expression. So, the new expression becomes \(2 (4)^{3}-3 (6)\)

Step 2: Simplifying the expression

Next, calculate the expression. The expression now becomes \(2*64-18\)

Step 3: Final calculation

Finally, perform the required operations to obtain the result which is \(128 - 18\)

Key Concepts

Substitution in Expressions

Substitution is a fundamental step when working with algebraic expressions. It's the process of replacing variables in an expression with specific values. In this exercise, we need to substitute the given values for the variables to move forward efficiently. For example, in the expression \(2s^{3}-3t\), the variables are \(s\) and \(t\). The values provided are \(s = 4\) and \(t = 6\). To substitute these into the expression, you replace every \(s\) with 4 and every \(t\) with 6. This transforms the expression into \(2(4)^{3} - 3(6)\). Substitution is crucial because it allows us to convert a generalized expression into one we can evaluate with specific numbers. This step prepares the expression for further simplification.

Simplifying Expressions

Simplifying an expression involves performing all the possible calculations, in a systematic order, to reduce it to its simplest form. After substitution, our expression becomes \(2(4)^{3} - 3 \times 6\). Here are the steps to simplify it:

• Calculate the exponent: \(4^{3} = 64\). This step involves multiplying four by itself three times, which equals 64.
• Multiply and simplify: Now substitute back into the expression, it becomes \(2 \times 64 - 18\).
• Perform multiplication: \(2 \times 64 = 128\).

By systematically addressing each part of the expression, you simplify it from a complex to a more manageable one. This makes the final calculation easier and helps avoid mistakes.

Order of Operations in Evaluations

To correctly solve mathematical expressions, especially in algebra, the order of operations is critical. This set of rules defines the correct sequence to apply different operations, ensuring accuracy.Typically, the order of operations is remembered by the mnemonic "PEMDAS," which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our example, \(2 \times 64 - 18\), the order directs us to first handle any exponents, such as \(4^{3}\). Then, we move to multiplication\(2 \times 64\) before performing the final subtraction \(128 - 18\). Following these rules means operations are done logically and accurately, leading us to the correct final answer of 110.