Question #3876

If a,b ЄR and b≠0 show that:
a. |a|= √a[sup]2[/sup]
b. |a/b|= |a|/|b|

Expert's answer

a. Show that |a|= √a^{2}

Recall that

|a|=a if a>=0

and

|a|=-a otherwise.

Let a>=0. Then

sqrt{a^2} = a = |a|

Suppose a<0. Then a=-b, where b>0, so |a|=b.

On the other hand,

√a^{2} = √(-b)^{2} = √b^{2} = b = |a|

b. Show that |a/b|= |a|/|b|

Consider the following cases:

1) Let a>=0, b>0.

Then |a/b|= a/b = |a|/|b|.

2) Suppose a>=0, b<0.

Then a/b<0, |a|=a, |b|=-b, so

|a/b|= -(a/b) = a/(-b) = |a|/|b|.

3) Suppose a<0, b>0.

Then a/b<0, |a|=-a, |b|=b, so

|a/b|= -(a/b) = (-a)/b = |a|/|b|.

3) Finally, let a<0, b<0.

Then a/b>0, |a|=-a, |b|=-b, so

|a/b|= a/b = (-a)/(-b) = |a|/|b|.

Recall that

|a|=a if a>=0

and

|a|=-a otherwise.

Let a>=0. Then

sqrt{a^2} = a = |a|

Suppose a<0. Then a=-b, where b>0, so |a|=b.

On the other hand,

√a

b. Show that |a/b|= |a|/|b|

Consider the following cases:

1) Let a>=0, b>0.

Then |a/b|= a/b = |a|/|b|.

2) Suppose a>=0, b<0.

Then a/b<0, |a|=a, |b|=-b, so

|a/b|= -(a/b) = a/(-b) = |a|/|b|.

3) Suppose a<0, b>0.

Then a/b<0, |a|=-a, |b|=b, so

|a/b|= -(a/b) = (-a)/b = |a|/|b|.

3) Finally, let a<0, b<0.

Then a/b>0, |a|=-a, |b|=-b, so

|a/b|= a/b = (-a)/(-b) = |a|/|b|.

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