Problem 44 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(9(-2)(-r)^{3}\) when \(r=2\)

Step-by-Step Solution

Answer

The evaluated output of the expression \(9(-2)(-r)^{3}\) when \(r=2\) is 144.

Step 1: Substitute the Given Value

Substitute the given value of 'r' into the expression. So, the expression \(9(-2)(-r)^{3}\) becomes \(9(-2)(-2)^{3}\).

Step 2: Apply the Power to the Variable

Solve \((-2)^{3}\) first according to the rule of precedence (BIDMAS/BODMAS). This results in \(-8\) . So we update our expression to \(9(-2)(-8)\).

Step 3: Complete the Multiplication

Complete the multiplication operation for the expression \(9(-2)(-8)\) which gives result as \(144\).

Key Concepts

Substitution

When evaluating mathematical expressions, we often first encounter the concept of **substitution**. Substitution involves replacing a variable in an expression with its given numerical value. In this case, the original expression is given as \(9(-2)(-r)^{3}\) and we know that \(r\) equals 2.

Here's how substitution works step-by-step:

• Identify the variable in the expression, which is \(r\) in our exercise.
• Replace the variable with its assigned numerical value. Since \(r = 2\), wherever there is \(-r\), substitute \(-2\).
• The expression now becomes \(9(-2)(-2)^{3}\). Each occurrence of \(r\) is replaced, simplifying further calculations.

Substitution is a crucial step because it sets the stage for accurate simplification of expressions. Always double-check your substituted values to prevent errors in later steps.

Exponents

An exponent refers to the number of times a number, known as the base, is multiplied by itself. In our example, the expression \((-r)^{3}\) features an exponent. This means we will multiply the base by itself three times.

Here's how it breaks down for \(-r = -2\), with the exponent of 3:

• Multiply \(-2\) by itself three times, which is \((-2) \times (-2) \times (-2)\).
• The first multiplication: \((-2) \times (-2) = 4\).
• The second multiplication: \(4 \times (-2) = -8\).

Remember, exponents follow specific rules:

• A negative number raised to an odd exponent, such as 3, remains negative.
• When you perform operations involving exponents, do so before proceeding with addition, subtraction, or multiplication outside the powers.

Accurately handling exponents is key to reaching the correct solution, as it directly affects the sign and magnitude of the result.

Order of Operations

The **order of operations** is a fundamental concept ensuring that expressions are evaluated consistently. In mathematics, the order is often summarized by the acronym BIDMAS/BODMAS. This stands for Brackets, Indices (another term for exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Let's consider our expression, \(9(-2)(-8)\):

• The original expression involves multiplication and an exponent, so according to BIDMAS, we handle the exponent first.
• After evaluating the exponent step as described earlier, we address the multiplication. Simplifying from left to right, multiply the constants in sequence:
• First, \(9 \times (-2) = -18\) because multiplication of a positive and negative number results in a negative product.
• Next, multiply \(-18 \times (-8) = 144\). Multiplying two negative numbers results in a positive number.

Following the order of operations ensures accuracy and consistent results, no matter how complex the expression. Adhering to these rules helps avoid mistakes that can lead to incorrect answers.