Problem 21 - Elementary and Intermediate Algebra

Question

Simplify. $$ 7^{1} $$

Step-by-Step Solution

Answer

7

Step 1: Understand the exponent

The exponent here is 1. Any number raised to the power of 1 remains the same.

Step 2: Apply the exponent rule

Use the rule for exponents which states that for any number a, \[ a^{1} = a \].

Step 3: Simplify the expression

Given \[ 7^{1} \], apply the rule: \[ 7^{1} = 7 \].

Key Concepts

Simplifying Exponents

Exponents are a way to express repeated multiplication of a number by itself. In this exercise, we were asked to simplify the expression \(7^{1} \). When dealing with exponents, it's important to understand and apply the basic exponent rules. An exponent tells us how many times to use the base as a factor.
For instance, in the expression \(7^{1}\), the base is 7 and the exponent is 1. Now, the rule for any number raised to the power of 1 is simple: \(a^{1} = a\). So, \(7^{1} = 7\).Understanding these basic exponent rules will make simplifying expressions much easier.

Basic Exponent Rules

Learning and remembering some fundamental exponent rules can greatly simplify the process of working with mathematical expressions involving exponents. Here are some essential rules:

• \(\textbf{Product of Powers Rule:} a^{m} \times a^{n} = a^{m+n}\)
• \(\textbf{Quotient of Powers Rule:} a^{m} \div a^{n} = a^{m-n}\)
• \(\textbf{Power of a Power Rule:} (a^{m})^{n} = a^{mn} \)
• \(\textbf{Zero Exponent Rule:} a^{0} = 1\) (given that a is not zero)
• \(\textbf{Negative Exponent Rule:} a^{-n} = \frac{1}{a^{n}}\)

Applying these rules can make complex algebraic expressions simpler and easier to manage. In our exercise, we used the rule that any number to the power of one remains the same number.

Algebra and Mathematical Expressions

Algebra involves working with symbols and letters to represent numbers in equations and formulas. Exponents are a key aspect of algebra, as they help in representing large numbers and complex calculations concisely.
When simplifying mathematical expressions with exponents, it's crucial to apply the appropriate rules correctly. These principles are foundational for solving more complicated algebraic problems.
For example, let's consider the expression \(a \times b^{2} \). Here, 'a' is a coefficient and \(b^{2}\) includes an exponent. Depending on the values of 'a' and 'b', and using exponent rules, this expression can be simplified or expanded to solve algebraic equations.
By mastering basic exponent rules and understanding their application, students can efficiently handle a wide range of algebraic problems.