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Chapter 8: Problem 12

Compute the exact value. $$ 3^{2} $$

Short Answer

Expert verified
The exact value of \(3^2\) is 9.

Step by step solution

01

Understand Exponentiation

The question requires us to compute the value of an expression using the exponentiation operation. Exponentiation involves raising a base number to the power given by the exponent.
02

Identify the Base and Exponent

For the expression \(3^2\), the base is 3 and the exponent is 2. This means we need to multiply the base by itself as many times as the exponent indicates.
03

Perform the Multiplication

To calculate \(3^2\), multiply 3 by itself: \(3 \times 3\).
04

Calculate the Result

Evaluate the multiplication \(3 \times 3\), which equals 9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
When we talk about exponents, we're dealing with a mathematical operation that involves two numbers: a base and an exponent. The exponent tells you how many times to multiply the base by itself. In simpler terms, it's a shorthand way of expressing repeated multiplication.
  • For example, in the expression \(3^2\), 3 is the base that you are multiplying.
  • The small number 2, positioned at the upper right, is the exponent.
Exponentiation is a fundamental concept in math, frequently used in areas such as algebra, calculus, and even in real-world applications like computing and finance.
Understanding how exponents function helps simplify calculations and solve complex problems more efficiently.
Base and Exponent
To delve deeper into understanding base and exponent, let's elaborate using the expression \(3^2\).
  • The base is 3, which is the number you repeatedly multiply.
  • The exponent is 2, which indicates how many times you'll multiply the base by itself.
In this particular example, the exponent 2 shows that the base 3 should be used as a factor twice: \(3 \times 3\). This process is known as squaring because you are finding the area of a square where each side measures 3 units.
Understanding the concept of base and exponent not only aids in simplifying calculations but it also prepares you for more complex operations involving powers, such as roots and logarithms.
Multiplication
Multiplication is a core arithmetic operation, and it's essential to understand it deeply, especially when dealing with exponents. In the expression \(3^2\), you multiply the base (3) by itself according to the exponent (2).
  • Starting with the base 3, multiply it by itself once, because the exponent is 2: \(3 \times 3\)
  • This results in 9, which is the answer to our exponentiation \(3^2\). The repeated multiplication here results in what's known as a power.
Multiplying in this context is straightforward but powerful as it quickly escalates numbers in size with higher exponents.
In arithmetic, multiplication is also the stepping stone to more intricate mathematical concepts, such as algebraic expressions and geometric measurements.

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Most popular questions from this chapter

Compute the exact value. $$ \left(\frac{1}{3}\right)^{-3} $$

Simplify the expression, and write your answer in the form \(x^{r}\). $$ \left(x^{\frac{1}{2}}\right)^{\frac{4}{3}} $$

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Simplify the quotient, and write your answer in the form \(x^{r}\). $$ \frac{x^{-\frac{5}{2}}}{x^{\frac{5}{2}}} $$
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