Prove if a belongs R, then (a^2)^1/2= |a|?

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In summary, for a number to belong to R, it must be a real number. The notation (a^2)^1/2 means to take the square root of a squared number, resulting in the absolute value of a. It is necessary to prove that (a^2)^1/2 equals |a| in order to show that it holds true for all real numbers a. The proof involves using algebraic manipulations and the properties of real numbers. Proving this statement has implications for solving equations involving square roots and absolute values, and helps to solidify our understanding of real number properties.
  • #1
hugo28
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Would you please help me? Thanks in advance.

Prove that if a belongs to R, then (a^2)^1/2 = |a|?
by using Abstract, Discrete, Algebraic Math.

I work, but stuck:
|a| =< (a^2)^1/2, then
- (a^2)=< a =< (a^2)

Please help!
 
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  • #2
Depends on your definition of square root. The usual definition means you will have two answers a and -a. However if you insist that the square root be positive, the |a| is the positive root.
 
  • #3
If a >= 0, it's pretty much obvious; otherwise, consider -a.
 
  • #4
Thread closed for user not showing any work.
 
  • #5


Abstractly, we can prove this statement by using the definition of absolute value and the properties of real numbers. The absolute value of a real number is defined as its distance from 0 on the number line, so it is always a positive value. In contrast, the square root of a positive number is also always positive. Therefore, if a belongs to R, then both |a| and (a^2)^1/2 are positive values.

Discretely, we can prove this statement by considering different cases for the value of a. If a is positive, then both |a| and (a^2)^1/2 will be equal to a. If a is negative, then |a| will be equal to -a and (a^2)^1/2 will be equal to -a as well, since the square of a negative number is positive. Therefore, in both cases, |a| will be equal to (a^2)^1/2.

Algebraically, we can prove this statement by using the properties of exponents. Since a belongs to R, we can write a as a product of its factors, where some of the factors may be repeated. For example, a = p*q*r*...*r, where p, q, and r are real numbers and * represents multiplication. Then, (a^2)^1/2 can be written as (p*q*r*...*r)^1/2 = (p^2*q^2*r^2*...*r^2)^1/2 = p*q*r*...*r = a. Similarly, |a| can be written as |p*q*r*...*r| = |p|*|q|*|r|*...*|r| = p*q*r*...*r = a. Therefore, (a^2)^1/2 and |a| are equal and the statement is proven.
 

1. What does it mean for a number to belong to R?

In mathematics, R refers to the set of all real numbers, which includes all rational and irrational numbers. Therefore, for a number to belong to R, it must be a real number.

2. What does the notation (a^2)^1/2 mean?

This notation means to take the square root of a squared number, which results in the absolute value of a. In other words, it is the positive square root of a^2.

3. Why is it necessary to prove that (a^2)^1/2 equals |a|?

In mathematics, it is important to prove statements or equations in order to show that they are always true, regardless of the value of the variables involved. In this case, proving that (a^2)^1/2 equals |a| means showing that it holds true for all real numbers a.

4. How can we prove that (a^2)^1/2 equals |a|?

The proof involves using the properties of real numbers and the definition of absolute value. We can start by assuming that a belongs to R, and then use algebraic manipulations to show that (a^2)^1/2 equals |a|.

5. What are the implications of proving that (a^2)^1/2 equals |a|?

Proving this statement has several implications. It shows that the square root of a squared number is always equal to the absolute value of that number, regardless of the value of a. This can be useful in solving equations involving square roots and absolute values, and it also helps to solidify our understanding of the properties of real numbers.

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