Problem 7 - Elementary Algebra

Question

For Problems \(1-20\), find the value of each numerical expression. For example, \(2^{4}=2 \cdot 2 \cdot 2 \cdot 2=16\). $$ -3^{2} $$

Step-by-Step Solution

Answer

The answer is \(-9\).

Step 1: Identify the Base and Exponent

The expression is \(-3^2\). The base number here is \(3\) (note it should be positive as per order of operations) and the exponent is \(2\). We are calculating \(3^2\) and then applying the negative sign as per BEDMAS/BODMAS rules.

Step 2: Calculate the Exponentiation

The exponentiation of \(-3^2\) is evaluated as \((3)^2\). So, compute \(3 \times 3\).This results in \(9\).

Step 3: Apply the Negative Sign

After calculating \(3^2\), we need to apply the negative sign since the original expression was \(-3^2\). Thus, we take \(-9\).

Key Concepts

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. In mathematical terms, if you have a base number, say 3, and it is raised to the power of 2, you write this as \(3^2\). This means you multiply 3 by itself: \(3 \times 3 = 9\). Exponents are an important part of math because they help simplify expressions and solve equations.
If you are dealing with other numbers, for example, \(a^4\), this notation tells you to multiply \(a\) four times: \(a \times a \times a \times a\).
Remember to perform exponentiation before moving on to other operations, according to the order of operations rules. This ensures accuracy in solving mathematical expressions.

Negative Numbers

Negative numbers are numbers that are less than zero and are often represented with a minus sign. In mathematical operations, treating negative numbers correctly is crucial.
When it comes to exponents and negative numbers like in the expression \(-3^2\), the placement of negative signs plays a key role. The expression \(-3^2\) is different from \((-3)^2\).

• In \(-3^2\), you first calculate the exponentiation \(3^2\) which is 9, and then apply the negative, resulting in -9.
• In \((-3)^2\), the negative sign is part of the base, meaning \((-3) \times (-3)\), which results in a positive 9.

Understanding these differences helps prevent mistakes in calculations.

BEDMAS/BODMAS

The terms BEDMAS and BODMAS refer to the order of operations in mathematical expressions. Understanding these is vital in solving expressions accurately.
BEDMAS stands for:

• **B**rackets first ([])
• **E**xponents (powers and roots, etc.)
• **D**ivision and **M**ultiplication (left-to-right)
• **A**ddition and **S**ubtraction (left-to-right)

BODMAS follows the same principle with ***O*** standing for **Order** (exponents and square roots).
When you see an expression, follow these steps to ensure you perform the operations in the correct order. For example, in handling \(-3^2\), you prioritize exponents. Calculate \(3^2\) first to get 9, then apply the negative sign as guided by BEDMAS/BODMAS, which results in -9. Keeping this order in mind helps resolve complex expressions without errors.