Which is larger, 5 + 2 4 5 or 5 × 2 4 5 ?
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If you're wondering why those particular numbers, note that 2 4 = 5 2 − 1 , and in general, for a = 0 or a > 1 ,
a + a 2 − 1 a = a a 2 − 1 a .
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?
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Actually, when one writes the later expression, the 5 is much smaller, like this: 5 2 4 5
Short and sweet solution.
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.
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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
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In the context of old typesetting it could be confused, but 5 2 4 5 is definitely different from 5 2 4 5 .
You are missing an add sign in A
Didn' t see the third option, for real: why would anyone even suggest to know algebra, if otherwise? I need help.
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Are you asking what motivates people to study algebra?
I decided to try it out. I didn't really get it. I should look at it more carefully
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You could ask the author for clarification if you did not get his solution.
Wow, thanks. That is really cool.
Relevant wiki: Square Roots
First one= 5 + 2 4 5 = 2 4 1 2 0 + 5 = 2 4 1 2 5
Second one= 5 2 4 5 = 2 5 × 2 4 5 = 2 4 5 × 2 5 = 2 4 1 2 5
so, First one=Second one.[they are equally same]
Can we simplify 2 4 1 2 5 to 1 2 5 3 0 ?
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obviously...... 2 4 1 2 5 = 2 × 2 × 2 × 3 5 × 5 × 5 = 2 6 5 5 = 2 6 6 5 5 6 = 1 2 5 3 0
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That does not make it obvious, though.
Relevant wiki: Square Roots
Suppose, x = 5 + 2 4 5 and y = 5 2 4 5
x = 5 + 2 4 5
= 5 + 2 4 5
= 1 5 + 2 4 5 Converting element to fraction
= 2 4 5 ⋅ 2 4 + 2 4 5 LCD of 1 5 + 2 4 5 is 2 4
= 2 4 5 ⋅ 2 4 + 5
= 2 4 1 2 5
____________________
y = 5 2 4 5
= 2 4 5 ⋅ 2 5
= 2 4 1 2 5
Hence 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.
Like Munems best.
The second one looks like the 5th root of that, not 5* the square root of that. :/
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Suppose both are equal
By squaring on both sides we get......
5 + 5/24 = 25 * 5/24
By solving........
125/24 = 125/24
Notice that, by the distributive property:
5 + 2 4 5 = 2 4 5 × ( 1 + 2 4 ) = 2 4 5 × 2 5 = 5 2 4 5
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.
Let's generalize for now by setting x=5 and y = 24, then
x
+
y
x
=? >=? <=?
x
y
x
Whether an equality or inequality we can square both sides (and only propagate the = signs for now, to simplify) then
x+
y
x
=
y
x
3
multiplying across by y and simplifying we get
xy + x =
x
3
y =
x
2
- 1
So the terms are equal for all xy, meeting the conditions above, which includes x=5 and y = 24.
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Relevant wiki: Square Roots
A = 5 2 4 5 = 2 4 2 4 × 5 + 5 = 2 4 1 2 5
B = 5 2 4 5 = 2 4 5 × 2 5 = 2 4 1 2 5
∴ A = B