What is the value of (sqrt(24 + 5 sqrt(23)))(sqrt(24 - 5 sqrt(23))?

Question

What is the value of ( \sqrt{24 + 5\sqrt{23}})(\sqrt{24 - 5 \sqrt{23}} )?

A. 48

B.   \sqrt{24}

C.  5 \sqrt{2}

D. 1

E.  24 - 5 \sqrt{23}

chetan2u · Accepted Answer

The moment you look at something of the form (a+b)(a-b), you are looking at formula of  a^2-b^2

\sqrt{24 + 5\sqrt{23}}*\sqrt{24 - 5 \sqrt{23}} 
 \sqrt{(24 + 5\sqrt{23})(24 - 5 \sqrt{23})} 
 \sqrt{24^2-(5 \sqrt{23})^2} 
 \sqrt{24*24-25*23} 
 \sqrt{576-575} 
 \sqrt{1} 
 1

D

BrushMyQuant · Answer

↧↧↧ Weekly Video Solution to the Problem Series ↧↧↧

https://www.youtube.com/watch?v=FY0yCp-Wyzw

We need to find the value of ( \sqrt{24 + 5\sqrt{23}})(\sqrt{24 - 5 \sqrt{23}} )

( \sqrt{24 + 5\sqrt{23}})(\sqrt{24 - 5 \sqrt{23}} ) =  \sqrt{(24 + 5\sqrt{23}) * (24 - 5 \sqrt{23})}

If we look at the expression inside the square root then we will notice that both the terms are similar but have different signs in the middle
=> They are of the form (a-b) * (a+b) and will be equal to  a^2 - b^2

=>  \sqrt{(24 + 5\sqrt{23}) * (24 - 5 \sqrt{23})}  =  \sqrt{(24^2 - (5\sqrt{23})^2))} 
=  \sqrt{576 - 25*23}  =  \sqrt{576 - 575}  =  \sqrt{1}  = 1

So,  Answer will be D 
Hope it helps!

Watch the following video to learn the Properties of Roots

https://www.youtube.com/watch?v=5oAec-Y6kOQ

TPW · Answer

We need to find the value of (24+5 ^1/2)^1/2 * (24-5 ^1/2)^1/2

If we look at the expression inside the square root then we will notice that both the terms are similar but have different signs in the middle
=> They are of the form (a-b) * (a+b) and will be equal to

So you can reach (24^2 - (5 ^1/2)^2)^1/2

Or
(24*24 - 25*23)^1/2

Here is where you can again save time and calculation if you notice the pattern that the second term is distributed around 24 and can be written as
(24^2- (24+1)*(24-1))^1/2

Which is nothing but
(24^2 - (24^2 - 1^2))^1/2

Or

(24^2-24^2+1^2)^1/2

= (1^2)^1/2 = 1 
Choice D I guess

Posted from my mobile device

ablatt4 · Answer

I incorrectly answer E, my mistake was simple but easy to make.

This is an example of a^2-b^2

sqrt 24^2- 5^2 (23)
*my mistake*

I wiped the roots with the exponents and answered E HOWEVER, this is not allowed in subtraction or additoin under roots.

What you should do is solve for 24*24 and 23*25 and get 576-575=1
All of this was still under the root and the root of 1 is 1 so D is correct.

ablatt4 · Answer

The first thing that is significant is this questions is a form of a^2-b^2.

Put everything under 1 root (you can do this with multiplication and division but not addition and subtraction)

translate this into sqrt 24^2- (23*25)^2

solve and get 576-575=1

D

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What is the value of (sqrt(24 + 5 sqrt(23)))(sqrt(24 - 5 sqrt(23))?

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What is the value of (sqrt(24 + 5 sqrt(23)))(sqrt(24 - 5 sqrt(23))?

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