Problem 83 - Elementary Algebra Within Reach

Question

Evaluate the expression. $$3^{2}-(-4)$$

Step-by-Step Solution

Answer

The evaluated expression equals 13

Step 1: Compute the square of the number 3

The operation \(3^2\) means multiply 3 by itself: \(3 * 3 = 9\)

Step 2: Handle negative numbers: subtracting a negative number

When you observe \(9 - (-4)\), remember that subtracting a negative is the same as adding a positive. This expression transforms into \(9 + 4\)

Step 3: Evaluate the final expression

Add 9 and 4 to obtain the final result: \(9 + 4 = 13\)

Key Concepts

Exponents

In mathematics, an exponent refers to the number of times a number, called the base, is multiplied by itself. An expression like \(3^2\) is read as 'three to the power of two' or more simply, 'three squared.' To evaluate it, the base (3) is multiplied by itself, resulting in \(3 * 3 = 9\).

Exponents can be more complex, including negative exponents and fractional exponents, but the principle remains the same: it's a compact way to express repeated multiplication. It's critical to pay attention to the exponent value since it dictates how many times you multiply the base by itself, which greatly impacts the final result. For instance, \(3^3\) would be \(3 * 3 * 3 = 27\) – a very different outcome than \(3^2\).

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They are the opposite of positive numbers and are necessary for expressing values below zero, such as temperatures or debt.

Subtracting a negative number is often a point of confusion. Remembering that 'two negatives make a positive' is a helpful way to approach this. When you come across \(9 - (-4)\), you are essentially adding 4 to 9 since subtracting a negative number is equivalent to adding its positive counterpart. This turns our example into \(9 + 4\), simplifying the process of finding the correct answer by avoiding confusion around the negative sign.

Order of Operations

The order of operations is a fundamental concept in arithmetic and algebra. It outlines the specific sequence in which different operations should be carried out in a mathematical expression to ensure that everyone gets the same answer. The standard rules, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction), dictate this sequence: solve what's in parentheses first, then exponents, followed by any multiplication and division from left to right, and finally any addition and subtraction from left to right.

Applying these rules, in an expression like \(3^2 - (-4)\), you would first compute the exponent \(3^2=9\), then address the subtraction of a negative number, which becomes addition \(9+4\), and lastly, perform the addition to reach the final result of 13.