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Chapter 0: Problem 121

Describe what it means to raise a number to a power. In your description, include a discussion of the difference between \(-5^{2}\) and \((-5)^{2}\)

Short Answer

Expert verified
Raising a number to a power refers to multiplying the base number by itself according to the number of times stated by the exponent. The difference between \(-5^{2}\) and \((-5)^{2}\) lies in the treatment of the negative sign. \(-5^{2}\) equals -25 since 'negative' is an operation with the base 5. \((-5)^{2}\) equals 25, as the base here is -5.

Step by step solution

01

Introduction to Power and its Properties

Starting with the concept of exponentiation, it involves two numbers, where one number is the base and the other is the exponent. When a number, say a, is raised to the power n, written as \(a^n\), it implies that the base 'a' is being multiplied by itself 'n' times.
02

Understanding Negative Signs and Powers

A negative sign can either belong to the base or it can be an operation with the base. If the negative sign belongs to the base and the base is raised to a power, we put the base in parentheses, like \((-a)^n\). If the negative sign is an operation with the base, then it should be written as \(-a^n\). Note that multiplication and division have higher precedence than addition and subtraction.
03

Analyze \(-5^{2}\) and \((-5)^{2}\)

In case of \(-5^{2}\), the negative sign is an operation with the base 5. Here, the square operation applies only to 5, not to -5. Hence, \(-5^{2}\) equals -25. Whereas, for \((-5)^{2}\), -5 as a whole is the base and it is squared. Hence, \((-5)^{2}\) equals 25.

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