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Magazine Chapter 1: Problem 55
Solve the given problems. Is it always true that \(\sqrt{a^{2}}=a ?\) Explain.
Short Answer
Expert verified
No, \( \sqrt{a^2} = a \) only if \( a \geq 0 \); otherwise, \( \sqrt{a^2} = |a| \).
Step by step solution
01
Understand the Square Root and Square Relationship
The expression \( \sqrt{a^2} \) means the square root of \( a^2 \). Squaring a number \( a \) (\( a^2 \)) and taking the square root should ideally return us to the original number \( a \), but there are conditions to consider.
02
Consider the Definition of Square Roots
By definition, the square root of a number \( x \) is the non-negative number \( y \) such that \( y^2 = x \). Thus, \( \sqrt{a^2} = |a| \), the absolute value of \( a \), not simply \( a \).
03
Evaluate Different Scenarios for Variable \( a \)
1. If \( a \geq 0 \), then \( |a| = a \). Therefore, \( \sqrt{a^2} = a. \)2. If \( a < 0 \), then \( |a| = -a \). In this case, \( \sqrt{a^2} eq a \) but rather \( \sqrt{a^2} = -a. \)
04
Conclude the Evaluation
From the analysis, it's evident that \( \sqrt{a^2} = a \) is not always true because it only holds when \( a \) is non-negative. Otherwise, it equals the absolute value of \( a \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Root
The square root is a fundamental concept in algebra, and it refers to a number that, when multiplied by itself, gives the original number. When dealing with the expression \( \sqrt{a^2} \), we're noticing how closely linked squaring and taking square roots are.
To "square" a number means multiplying that number by itself: \( a \times a = a^2 \).
In contrast, the "square root" is about finding a number that can be squared to get the original value. In most mathematical contexts, the square root is defined to be the non-negative root.
To "square" a number means multiplying that number by itself: \( a \times a = a^2 \).
In contrast, the "square root" is about finding a number that can be squared to get the original value. In most mathematical contexts, the square root is defined to be the non-negative root.
- For positive numbers \( a \), \( \sqrt{a^2} = a \).
- For negative numbers \( a \), because we're looking for a non-negative result, \( \sqrt{a^2} = |a| \).
Absolute Value
The concept of absolute value is vital when discussing expressions like \( \sqrt{a^2} \). The absolute value of a number denotes its distance from zero on the number line, regardless of direction. It is always a non-negative number.
Symbolically, it is written as \( |a| \). Understanding the absolute value involves realizing:
This why the real solution to \( \sqrt{a^2} \) is \( |a| \), enforcing the idea that even if \( a \) is negative, \( \sqrt{a^2} \) becomes positive.
Symbolically, it is written as \( |a| \). Understanding the absolute value involves realizing:
- \( |a| = a \) if \( a \geq 0 \).
- \( |a| = -a \) if \( a < 0 \).
This why the real solution to \( \sqrt{a^2} \) is \( |a| \), enforcing the idea that even if \( a \) is negative, \( \sqrt{a^2} \) becomes positive.
Mathematical Proof
A mathematical proof provides a rigorous argument to confirm the validity of a particular statement. Inquiring whether \( \sqrt{a^2} = a \) is always true requires careful consideration backed by proof. To prove such statements, we need to cover all possible scenarios.
Here is a simplified approach:
Thus, the proof leads us to conclude that while \( \sqrt{a^2} = a \) might seem intuitive, the truthfulness hinges on the positivity of \( a \).
Here is a simplified approach:
- If \( a \geq 0 \), then indeed \( \sqrt{a^2} = a \) because both sides are naturally non-negative.
- However, if \( a < 0 \), \( \sqrt{a^2} = -a \), hence \( \sqrt{a^2} = a \) does not hold.
Thus, the proof leads us to conclude that while \( \sqrt{a^2} = a \) might seem intuitive, the truthfulness hinges on the positivity of \( a \).
Number Properties
Understanding number properties is crucial for expert handling of algebraic expressions. Several properties help elucidate relations between numbers.
Furthermore, recognizing that negating a negative restores positivity is foundational in comprehending why \( \sqrt{a^2} = |a| \). Through these properties, algebra becomes not only approachable but enjoyable.
- Commutative Property: The order of addition or multiplication doesn't change the result, e.g., \( a + b = b + a \).
- Associative Property: The grouping doesn't affect the sum or product, i.e., \( (a + b) + c = a + (b + c) \).
- Distributive Property: Allows multiplication to be distributed over addition, e.g., \( a(b + c) = ab + ac \).
Furthermore, recognizing that negating a negative restores positivity is foundational in comprehending why \( \sqrt{a^2} = |a| \). Through these properties, algebra becomes not only approachable but enjoyable.
