SOLUTION: sqrt(2sqrt21+22) - sqrt21 = 1 explain this clearly pls

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Question 1044700: sqrt(2sqrt21+22) - sqrt21 = 1 explain this clearly pls
Found 4 solutions by advanced_Learner, Theo, robertb, ikleyn:
Answer by advanced_Learner(501)   (Show Source):
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Answer by Theo(13342)   (Show Source):
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start with sqrt(2 * sqrt(21) + 22) - sqrt(21) = 1

add sqrt(21) to both sides of the equation to get:

sqrt(2 * sqrt(21) + 22) = 1 + sqrt(21)

square both sides of this equation to get:

2 * sqrt(21) + 22 = 1 + 2 * sqrt(21) + 21

simplify to get 2 * sqrt(21) + 22 = 22 + 2 * sqrt(21)

this can also be written as:

2 * sqrt(21) + 22 = 2 * sqrt(21) + 22

the equation is true which proves that the original equation is true.

you can also use your calculator to confirm.

in your scientific calculator, enter:

sqrt(2 * sqrt(21) + 22) - sqrt(21)

you will see that the result is equal to 1.

if your calculator doesn't do square roots, you can alternatively enter the expression as:

(2 * 21^(1/2) + 22)^(1/2) - 21^(1/2)

the result will also be equal to 1.

note that (1 + sqrt(21))^2 is equal to:

(1 + sqrt(21) * (1 + sqrt(21) which is equal to:

1 * 1 + 1 * sqrt(21) + sqrt(21) * 1 + sqrt(21) * sqrt(21) which is equal to:

1 + sqrt(21) + sqrt(21) + 21 which is equal to:

1 + 2 * sqrt(21) + 21 which is equal to:

22 + sqrt(21).

this is through use of the distributive law of multiplication.

Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website!

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Answer by ikleyn(52818)   (Show Source):
You can put this solution on YOUR website!
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For many other similar problems see the lessons
    - Calculations of expressions containing square roots
    - Prove that two irrational numbers are equal
    - Is this number rational or irrational?
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.