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Chapter 8: Problem 9

Compute the exact value. $$ 7^{1} $$

Short Answer

Expert verified
The exact value of \(7^1\) is 7.

Step by step solution

01

Identify the Base and Exponent

The expression given is \(7^1\). Here, 7 is the base and 1 is the exponent.
02

Understand Exponentiation

The exponent tells you how many times to multiply the base by itself. Any number raised to the power of 1 is simply the number itself, as you are not multiplying it by itself any additional times.
03

Compute the Value

Since the exponent is 1, you just take the base as it is. Thus, \(7^1 = 7\).

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Key Concepts

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

Base and Exponent
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number that is being multiplied. In the expression \(7^1\), the base is 7. The exponent, on the other hand, tells us how many times to multiply the base by itself. It is a small number written at the top right corner of the base. In this particular expression, the exponent is 1.
  • Base: The principal number being multiplied.
  • Exponent: Indicates the number of times the base is multiplied by itself.
These basic definitions are essential for understanding exponential expressions in mathematics. Whenever you see an expression like \(a^n\), remember: "a" is your base and "n" is your exponent.
Compute Value
Computing the value of an exponential expression involves translating the base and exponent into actual multiplication. However, any number raised to the power of 1 is simply itself. This is because the exponent tells us to multiply the base by itself only once.
For example:
  • \(7^1 = 7\) 鈥� Since the exponent is 1, you do not multiply 7 by anything else. It remains 7.
  • \(5^1 = 5\) 鈥� Again, the base remains unchanged.
This rule is a fundamental principle in exponentiation, making it intuitive for anyone learning basic algebra.
Intermediate Algebra
Intermediate algebra involves understanding more complex expressions and operations, of which exponentiation is a key concept. It builds upon the foundational idea of base and exponent to explore a variety of algebraic expressions. When you comprehend why \(7^1 = 7\), you're grasping a cornerstone that applies to broader algebraic operations.
  • Understanding exponents can help break down complex polynomial expressions.
  • Aids in solving equations involving exponential growth or decay.
By mastering these concepts, you're well on your way to tackling more sophisticated algebraic problems and concepts encountered in intermediate algebra.

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