Problem 12 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when \(x=3.\) $$ x^{3}+5 x $$

Step-by-Step Solution

Answer

The variable expression \(x^{3}+5x\) evaluates to \(42\) when \(x=3\).

Step 1: Substitute the given value

First, substitute the value of \(x\), which is given as \(3\), in the expression. This will give you \(3^{3} + 5*3\).

Step 2: Apply BODMAS/BIDMAS

Next, by following the BODMAS/BIDMAS rule which dictates the order of mathematical operations, raise \(3\) to power \(3\) and multiply \(5\) by \(3\). This will give you \(27 + 15\).

Step 3: Final Addition

Finally add the two numbers \(27 + 15\) together. This will yield the value of \(42\).

Key Concepts

Variable Substitution

Imagine having a jigsaw puzzle where each piece fits perfectly into place. Substitute the variables in an algebraic expression like filling in those puzzle pieces. It's the step of replacing variables with values.
In the given exercise, the variable is \(x\), and you're asked to find the expression’s value when \(x = 3\). So your first move is to replace every occurrence of \(x\) in the expression \(x^3 + 5x\) with \(3\).
This transforms your expression into \(3^3 + 5 \times 3\).

• This step makes your expression numbers-only, simplifying it for further operations.

Substitution is about putting values where variables are to see how the expression behaves or what it calculates to.

BODMAS/BIDMAS

Think of BODMAS/BIDMAS as a helpful checklist. It ensures you always perform mathematical operations in the correct sequence.
Here's a quick reminder of what BODMAS stands for:

• B – Brackets first
• O – Orders (i.e., powers and square roots, etc.)
• D – Division
• M – Multiplication
• A – Addition
• S – Subtraction

Following this order avoids confusion.
In our example, after substituting \(x = 3\) in \(x^3 + 5x\), the expression became \(3^3 + 5 \times 3\).

Applying BODMAS/BIDMAS, the powers (or "Orders") comes first:

• Calculate \(3^3\) which is \(27\).
• Then, handle the multiplication: \(5 \times 3\) equals \(15\).
• Finally, it's time for the addition: Add \(27 + 15\) to get \(42\).

By sticking to BODMAS rules, computations become easier and mistakes are avoided.

Exponents

Exponents might sound fancy, but they're simpler than you think.
It's just repeated multiplication. For example, \(3^3\) means multiplying \(3\) by itself three times:

• \(3 \times 3 \times 3 = 27\)

Exponents show how many times we should use a number in a multiplication.
This superscript number ("the power") tells us this.
In the expression \(x^3 + 5x\), the exponent was \(3\), telling us to multiply \(3\) by itself three times.
Understanding exponents is valuable, as they simplify large multiplications into concise forms, making math tasks less daunting.

Evaluation

Evaluation is like solving a mystery where every clue leads to the answer. It involves calculating the numerical value of an expression after substitution.
So, once your algebraic expression is turned into numbers through substitution, you can evaluate it by following the arithmetic steps.
In our exercise, after variable substitution and applying BODMAS, you had these numbers: \(27 + 15\).

• By simply adding these two, you evaluated, or found, the expression's final value, which was \(42\).

Evaluation reveals the expression's value, giving you the "solution" to the math "puzzle."
Each calculation step, from substitution to applying rules, leads us towards this final answer. This process helps improve how we understand and solve algebraic problems.