Problem 14 - Beginning and Intermediate Algebra

Question

Evaluate. \(9^{2}\)

Step-by-Step Solution

Answer

The value of \(9^{2}\) is 81.

Step 1: Identify the Base and Exponent

In the expression \(9^{2}\), the number \(9\) is the base, and the number \(2\) is the exponent. The exponent tells us how many times we need to multiply the base by itself.

Step 2: Read the Definition of Exponents

When a number is raised to an exponent, it means you multiply the base (in this case, \(9\)) by itself as many times as indicated by the exponent. Here \(9^{2}\) means 9 multiplied by itself once.

Step 3: Multiply the Base by Itself

Carry out the multiplication of the base by itself: \(9 \times 9\). First, set up this simple multiplication equation.

Step 4: Perform the Calculation

Calculate the product: \(9 \times 9 = 81\). This is the result of multiplying the base by itself as many times as indicated by the exponent.

Key Concepts

Base and Exponent Identification

In mathematical expressions involving exponents, it is crucial to identify the parts correctly. In an expression like \(9^{2}\), each component has a specific role. The number \(9\) is known as the **base**. The base is the number that is being multiplied by itself.
The small number \(2\) positioned to the top right of the base is called the **exponent**. The exponent tells us how many times the base should be used in a multiplication. In this case, the exponent is \(2\), which means we will multiply the base (\(9\)) by itself once.
For expressions like \(a^b\), **a** stands for the base, and **b** is the exponent. This understanding is fundamental when evaluating expressions with exponents.

Multiplication

Multiplication is a core mathematical operation where numbers (called factors) are added together a specific number of times. In the context of exponents, multiplication is used repeatedly according to the instructions provided by the exponent.
For \(9^{2}\), we use multiplication to combine the factors. Since the exponent is \(2\), the multiplication is set up as \(9 \times 9\). Here, the base \(9\) is used as a factor two times.
It's helpful to recall some basic multiplication facts to make calculations quicker. For instance, knowing that \(9 \times 9\) equals \(81\) helps us get to the solution fast. Reading simple multiplication as repeated addition can sometimes aid in better comprehension: here, we are essentially adding \(9 + 9\) many times.

Exponentiation

Exponentiation is the process of raising a base to the power of an exponent. It describes how many times a number, known as the base, is multiplied by itself. For \(9^{2}\), **exponentiation** means \(9\) is multiplied by itself according to the power, which is \(2\).
The exponent \(2\) in \(9^{2}\) indicates that \(9\) is repeated in the multiplication process two times (i.e., \(9 \times 9\)). The operation could be interpreted as finding the area of a square with side length \(9\), another practical interpretation of exponentiation in geometry.
Exponents are used in various fields beyond basic mathematics, including physics and computer science, to simplify expressions and calculations involving large numbers. Recognizing the pattern in exponentiation helps in solving more complex equations and understanding growth models.