Loving Derrick Niederman's books 1

Algebra Level 2

Which is larger, 5 + 5 24 or 5 × 5 24 ? \sqrt{ 5 + \frac5{24} }\quad \text{ or }\quad 5 \times \sqrt{\frac5{24} }\,?

5 + 5 24 \sqrt{ 5 + \dfrac5{24} } 5 5 24 5\sqrt{ \dfrac5{24} } They are equal

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

7 solutions

Tommy Li
Aug 29, 2017

Relevant wiki: Square Roots

A = 5 5 24 = 24 × 5 + 5 24 = 125 24 \large A = \sqrt{5\frac{5}{24} } = \sqrt{\frac{24\times5+5}{24}} = \sqrt{\frac{125}{24}}

B = 5 5 24 = 5 × 25 24 = 125 24 \large B = 5\sqrt{\frac{5}{24}} = \sqrt{\frac{5\times 25}{24}} = \sqrt{\frac{125}{24}}

A = B \large ∴ A =B

Moderator note:

If you're wondering why those particular numbers, note that 24 = 5 2 1 , 24 = 5^2 - 1, and in general, for a = 0 a = 0 or a > 1 , a >1 ,

a + a a 2 1 = a a a 2 1 . \sqrt{ a + \frac{a}{a^2-1} } = a\sqrt{\frac{a}{a^2-1} }.

The proof is similar to various answers given for this problem, but with algebraic terms substituted in.

I read this not as B= 5 x sqrt( 5 / 24} but as B= ( 5 / 24 )^(1 / 5). My fault, maybe?

Ed Sirett - 3 years, 9 months ago

Log in to reply

Actually, when one writes the later expression, the 5 is much smaller, like this: 5 24 5 \sqrt[5]{\frac{5}{24}}

Agnishom Chattopadhyay - 3 years, 8 months ago

Short and sweet solution.

Thomas Sutcliffe - 3 years, 9 months ago

I'd agree with Ed on this - reading that looked as if it were the 5th root of 5/24 and that would make both of the other two answers correct depending on which root is chosen.

Craig Mitchell - 3 years, 9 months ago

Log in to reply

Additionally even if you follow the logic given it fails to consider the other root -5(5/24)^1/2 which definitely does not equal 5(5/24)^1/2

Craig Mitchell - 3 years, 9 months ago

Log in to reply

In the context of old typesetting it could be confused, but 5 5 24 5\sqrt{\frac{5}{24}} is definitely different from 5 24 5 \sqrt[5]{\frac{5}{24}} .

Brian Moehring - 3 years, 9 months ago

You are missing an add sign in A

A Former Brilliant Member - 3 years, 9 months ago

Log in to reply

Brilliant!

Andrea Cvečić - 3 years, 6 months ago

Didn' t see the third option, for real: why would anyone even suggest to know algebra, if otherwise? I need help.

Andrea Cvečić - 3 years, 8 months ago

Log in to reply

Are you asking what motivates people to study algebra?

Agnishom Chattopadhyay - 3 years, 8 months ago

I decided to try it out. I didn't really get it. I should look at it more carefully

Lucia Tiberio - 3 years, 8 months ago

Log in to reply

You could ask the author for clarification if you did not get his solution.

Agnishom Chattopadhyay - 3 years, 8 months ago

Wow, thanks. That is really cool.

James Wilson - 3 years, 7 months ago
Mohammad Khaza
Sep 10, 2017

Relevant wiki: Square Roots

First one= 5 + 5 24 \sqrt{5+\frac{5}{24}} = 120 + 5 24 \sqrt\frac{120+5}{24} = 125 24 \sqrt\frac{125}{24}

Second one= 5 5 24 5\sqrt\frac{5}{24} = 25 × 5 24 \sqrt{25} \times \sqrt\frac{5}{24} = 5 × 25 24 \sqrt\frac{5 \times 25}{24} = 125 24 \sqrt\frac{125}{24}

so, First one=Second one.[they are equally same]

Can we simplify 125 24 \sqrt{\dfrac{125}{24}} to 5 30 12 \dfrac{5 \sqrt{30}}{12} ?

Munem Shahriar - 3 years, 9 months ago

Log in to reply

obviously...... 125 24 \sqrt\frac{125}{24} = 5 × 5 × 5 2 × 2 × 2 × 3 \sqrt\frac{5 \times 5 \times 5}{2 \times 2 \times 2 \times 3} = 5 5 2 6 \frac{5\sqrt5}{2 \sqrt 6} = 5 5 6 2 6 6 \frac{5 \sqrt5 \sqrt6}{2 \sqrt6 \sqrt6} = 5 3 0 12 \frac{5 \sqrt30}{12}

Mohammad Khaza - 3 years, 9 months ago

Log in to reply

That does not make it obvious, though.

Agnishom Chattopadhyay - 3 years, 8 months ago
Munem Shahriar
Sep 10, 2017

Relevant wiki: Square Roots

Suppose, x = 5 + 5 24 x = \sqrt{5+ \dfrac{5}{24}} and y = 5 5 24 y = 5\sqrt{\dfrac{5}{24}}

x = 5 + 5 24 x = \sqrt{5+ \dfrac{5}{24}}

= 5 + 5 24 = \sqrt{5 + \dfrac{5}{24}}

= 5 1 + 5 24 = \sqrt{\dfrac{5}{1} + \dfrac{5}{24}} ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~~ Converting element to fraction \text{Converting element to fraction}

= 5 24 24 + 5 24 = \sqrt{\dfrac{5\cdot 24}{24} + \dfrac{5}{24}} ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~~ LCD of 5 1 + 5 24 is 24 \text{LCD of} \dfrac{5}{1} + \dfrac{5}{24} \text{is} ~ 24

= 5 24 + 5 24 = \sqrt{\dfrac{5 \cdot 24 +5}{24}}

= 125 24 = \sqrt{\dfrac{125}{24}}

____________________ \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}

y = 5 5 24 y = 5 \sqrt{\dfrac{5}{24}}

= 5 25 24 = \sqrt{\dfrac{5\cdot 25 }{24}}

= 125 24 = \sqrt{\dfrac{125}{24}}

Hence x = y \color{#3D99F6} \boxed {x = y}

That second expression looks like "the fifth root of " not "5 times". I'm sure a lot of people are getting it wrong for that reason.

Casey Hubbell - 3 years, 8 months ago

Like Munems best.

Brad Robinson - 3 years, 9 months ago

The second one looks like the 5th root of that, not 5* the square root of that. :/

Mac Was - 3 years, 8 months ago

Log in to reply

Yes, it should be phrased like this:

5 5 24 5 \cdot \sqrt{\dfrac{5}{24}}

Munem Shahriar - 3 years, 8 months ago
Azadali Jivani
Sep 11, 2017

Suppose both are equal
By squaring on both sides we get......
5 + 5/24 = 25 * 5/24
By solving........
125/24 = 125/24



Gabriel Cevanna
Sep 16, 2017

Notice that, by the distributive property:

5 + 5 24 = 5 24 × ( 1 + 24 ) = 5 24 × 25 = 5 5 24 \boxed{\sqrt{5+\frac{5}{24}}}= \sqrt{\frac{5}{24}\times(1+24)}= \sqrt{\frac{5}{24}}\times\sqrt{25}= \boxed{5\sqrt{\frac{5}{24}}}

Amaan Khan
Sep 15, 2017

If you're wondering why those particular numbers, note that and in general, for or

The proof is similar to various answers given for this problem, but with algebraic terms substituted in.

Bruce Neiger
Sep 11, 2017

Let's generalize for now by setting x=5 and y = 24, then
x + x y \sqrt{x+\frac{x}{y}} =? >=? <=? x x y x\sqrt\frac{x}{y}
Whether an equality or inequality we can square both sides (and only propagate the = signs for now, to simplify) then
x+ x y \frac{x}{y} = x 3 y \frac{x^3}{y} multiplying across by y and simplifying we get
xy + x = x 3 x^3
y = x 2 x^2 - 1
So the terms are equal for all xy, meeting the conditions above, which includes x=5 and y = 24.



0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...