Problem 54 Explain why $$ \left|a^{2}\r... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 0: Problem 54

Explain why $$ \left|a^{2}\right|=a^{2} $$ for every real number $a$.

Short Answer

Expert verified

In summary, the absolute value of $a^2$ is equal to $a^2$ for every real number $a$, because when $a$ is a positive or negative real number, squaring it results in a positive value, and its absolute value remains the same. When $a$ is zero, the result of squaring and its absolute value are also both zero.

Step by step solution

Case 1: Positive or Negative Real Number

If a real number $a$ is positive or negative, then we can write it as: $a > 0$ (Positive) or $a < 0$ (Negative) When we square a positive or negative real number, the result will always be positive since the product of two positive or negative numbers is always positive. Therefore: $a^2 > 0$ Since $a^2$ is positive, its absolute value will remain the same: $\left|a^{2}\right| = a^{2}$

Case 2: Real Number is Zero

If a real number $a$ is zero, then: $a = 0$ When we square a zero real number, the result will also be zero: $a^2 = 0$ Since $a^2$ is zero, its absolute value will also be equal to zero: $\left|a^{2}\right| = a^{2}$ Considering both cases, we have demonstrated that the absolute value of $a^2$ is equal to $a^2$ for every real number $a$.

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91影视

Explain why $$ \left|a^{2}\right|=a^{2} $$ for every real number \(a\).

Short Answer

Step by step solution

Case 1: Positive or Negative Real Number

Case 2: Real Number is Zero

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