Problem 81 - Prealgebra

Question

Perform the indicated operations. \(6^{2}\)

Step-by-Step Solution

Answer

The result of \(6^2\) is 36.

Step 1: Identify the operation

The expression given is \(6^2\). This indicates an exponentiation operation where 6 is the base and 2 is the exponent.

Step 2: Understand the exponentiation

Exponentiation means raising the base to the power of the exponent. In this case, \(6^2\) means multiplying 6 by itself once.

Step 3: Calculate the result of exponentiation

Compute \(6 imes 6\) to find the result. This yields \(36\).

Step 4: Conclude with the result

Conclude that \(6^2\) equals \(36\).

Key Concepts

Base and Exponent

In mathematics, exponentiation is a fundamental concept that deals with two key components: the base and the exponent. The base is the number we want to multiply multiple times, and the exponent tells us how many times to perform this multiplication.
For example, in the expression \(6^2\), the number 6 is the base, and the number 2 is the exponent. This notation is a way to compactly represent repeated multiplication. Instead of writing \(6 \times 6\), we write \(6^2\), which quickly communicates the operation being performed.
The base does not have to be a positive integer. It can be a negative number, a fraction, or even a variable, making exponentiation highly versatile in various mathematical problems. The exponent is usually a positive integer, but it can also be zero or negative, each with specific meanings and implications we will explore further in related topics.

Power of a Number

Understanding the concept of the power of a number is key to mastering exponentiation. When we raise a number to a power, we are essentially performing a specific type of repeated multiplication.
The power is denoted in the expression \(b^n\), where \(b\) is the base and \(n\) is the exponent, or the power. This means we multiply the base, \(b\), by itself \(n\) times. For instance, \(6^2\) is equivalent to \(6 \times 6\), simplifying to 36.
It's crucial to note:

• If the exponent is 1, the power is the base itself (e.g., \(6^1 = 6\)).
• If a number is raised to the power of 0, the result is always 1, assuming the base is not zero (e.g., \(6^0 = 1\)).
• For a negative power, the result is the reciprocal of the base raised to the positive power (e.g., \(6^{-2} = \frac{1}{6^2}\)).

This comprehension enables learners to tackle various algebraic and real-world problems efficiently.

Mathematical Operations

Mathematical operations are actions we can perform on numbers. Exponentiation, represented by a base raised to an exponent, is one of these operations. It is as essential as addition, subtraction, multiplication, and division.
When we encounter an expression like \(6^2\), understanding it as a mathematical operation is crucial. Here, the exponentiation operation is transforming the number 6. By multiplying it by itself, we simplify the expression to the result, 36.
The result, however, doesn't only depend on the base and exponent. Other operations can interact with exponentiation, like:

• Addition and subtraction, where results of exponentiated terms are summed or subtracted.
• Multiplication and division of powers with the same base or using the properties of exponents.
• More complex operations in algebra, involving brackets and exponents of sums or products.

These interactions highlight the importance of understanding the process thoroughly. Being fluent with exponentiation paves the way to mastering complex problems, fostering a deeper appreciation of mathematics.