Problem 103 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

Raising a number to a power means multiplying the number by itself multiple times. The quantity \(-5^{2}\) equals \(-25\), and \((-5)^{2}\) equals \(25\). Parentheses make all the difference - in the first case only 5 is squared and the negative is left as it is, in the second both -5 as a whole gets squared, hence resulting in a positive number.

Step 1: Understanding Power

Raising a number to a power means multiplying that number by itself certain number of times. The number being multiplied is the 'base' and the number of times is the 'exponent' or 'power'. For example, \(3^{2}\) means that we multiply 3 by itself twice, which equals 9.

Step 2: Negative Number to Power Without Parentheses

Without parentheses, the negative sign belongs only to the base and not to the whole expression. Therefore, \(-52\) would calculate as the negative of \(5^{2}\) as just the base is squared leaving the negative sign unaffected. Therefore, \(-5^{2}\) equals \(-25\).

Step 3: Negative Number to Power With Parentheses

With parentheses, the exponent applies to everything inside, including the negative sign. Therefore, \((-5)^{2}\) would calculate as \(25\), because squaring (multiplying a number by itself) a negative number results in a positive number.