Germany | A Nice Olympiad Algebra Problem

Ajouté : 2026-06-01

[00:00:00]Everyone, you're welcome to solve this nice algebra problem, which is the square root of x divided by x. This is equal to five.

[00:00:10]So, what is the value of x?

[00:00:12]Now, let's provide a solution from here.

[00:00:18]So, we have the square root of x divided by x. This is equal to five.

[00:00:26]Now, we have that the square root of a This is the same thing as a raised to the power of a half.

[00:00:34]Now, applying this property, this follows that we can express square root of x. This is the same thing as x raised to the power of a half divided by x. This is x raised to the power of one.

[00:00:47]Then, this is equal to five.

[00:00:52]The next step is that x to the power of a half divided by x to the power of one.

[00:00:57]This is in the form of a to the power of n divided by a to the power of m, which we can express as a to the power of n minus m.

[00:01:09]Now, let's apply this exponent property so that we have x raised to the power of a half subtract one.

[00:01:18]This is equal to five.

[00:01:23]Now, x to the power of a half minus one.

[00:01:26]This is the same thing as x raised to the power of negative a half.

[00:01:31]This is equal to five.

[00:01:36]Now, the next step is that x to the power of minus a half. This is in the form of a raised to the power of minus n, which we can express as one over a to the power of n.

[00:01:48]Applying this property, this implies that we have one over x raised to the power of a half.

[00:01:57]This is equal to five. This is equal to five.

[00:02:02]Now five is a whole number, so this is over one.

[00:02:06]The next step is to cross multiply from here. So let's cross multiply.

[00:02:13]So the next step is to cross multiply from here. So we have one times one.

[00:02:18]This is equal to five multiplying by X raised to the power of half.

[00:02:26]Now we have one. This is equal to five multiplying by X raised to the power of half.

[00:02:34]The next step from here, let's square on both sides. So let's square on both sides from here.

[00:02:42]Now one squared this is one. This is equal to five multiplying by X raised to the power of half, then raised to the power of two.

[00:02:54]We have that five times X to the X raised to the power of half raised to the power of two. This is in the form of A times B raised to the power of N, which you can express as A to the power of N multiplying by B to the power of N.

[00:03:10]Now applying this exponent property, this implies we have one. This is equal to five raised to the power of two multiplying by X raised to the power of half, then raised to the power of two.

[00:03:27]So we have one. This is equal to five squared. This is 25.

[00:03:33]Multiplying by X to the power of half raised to the power of two. This means we eliminate. This is two and two simplifies. So we have 25 times X.

[00:03:44]And this implies that we have one. This is equal to 25 X.

[00:03:51]So let's divide both sides by 25.

[00:03:55]And also here by 25.

[00:03:57]If we simplify here, this implies that X is equal to 1 over 25.

[00:04:05]This is equal to 1 over 25.

[00:04:09]The next step from here is to verify. So let's verify that this value of X here satisfies the equation.

[00:04:18]Now, if you recall, we have that the square root of X divided by X, this should give us a value of 5. This should give us a value 5.

[00:04:34]Now let's substitute the value of X here so that we have this is equal to the square root of X, which is 1 over 25 divided by 1 over 25.

[00:04:48]This is supposed to give us a value of 5.

[00:04:54]So the square root of 1 over 25, this is the same thing as 1 over 5 divided by 1 over 25.

[00:05:05]This should give us a value of 5.

[00:05:10]So we have here, this is 1 over 5 multiplying by 25 over 1.

[00:05:17]This is supposed to give us a value of 5.

[00:05:21]So we have here, let's simplify 25 divided by 5.

[00:05:25]This is equal to 5. So we have that 5 is equal to 5.

[00:05:31]So the left hand side is equal to the right hand side.

[00:05:38]And this proves that the value of X, which is 1 over 25, satisfies the equation.

[00:05:45]So kindly follow the steps.

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