methods for finding a surd expression for the positive square root of 7-4*sqrt[3]

[grant]

Hi all,

I think this may just scrape into the intermediate algebra category.

I am looking at modulus questions and have found an example which I do not understand. The example is as follows:

"Find a simple surd expression for the positive square root of 7-4*sqrt[3].
7-4*sqrt[3] = (2 - sqrt[3])^2 or equivalently (sqrt[3]-2)^2. However, sqrt[ 7-4*sqrt[3]] = 2 - sqrt[3] not sqrt[3]-2 since by definition it must be positive."

The bit i do not understand is how they computed that 7-4*sqrt[3] = (2 - sqrt[3])^2. the rest makes sense to me.

I can see that both (2-sqrt[3])^2 and (sqrt[3]-2) equal 7-4*sqrt[3], however no matter how i manipulate it, I just can't see how they could have reasoned that it did.

I figure that there must be some method of manipulating surd expressions of which I am unaware. I know basic algebra with regards to radical expressions.

I don't even really know what to look up to find out what this type of thing is (finding two termed surd expressions?). Can anyone provide me with some pointers on what is going on here and p

 

[Deleted member 4993]

Hi all,

I think this may just scrape into the intermediate algebra category.

I am looking at modulus questions and have found an example which I do not understand. The example is as follows:

"Find a simple surd expression for the positive square root of 7-4*sqrt[3].
7-4*sqrt[3] = (2 - sqrt[3])^2 or equivalently (sqrt[3]-2)^2. However, sqrt[ 7-4*sqrt[3]] = 2 - sqrt[3] not sqrt[3]-2 since by definition it must be positive."

The bit i do not understand is how they computed that 7-4*sqrt[3] = (2 - sqrt[3])^2. the rest makes sense to me.

I can see that both (2-sqrt[3])^2 and (sqrt[3]-2) equal 7-4*sqrt[3], however no matter how i manipulate it, I just can't see how they could have reasoned that it did.

I figure that there must be some method of manipulating surd expressions of which I am unaware. I know basic algebra with regards to radical expressions.

I don't even really know what to look up to find out what this type of thing is (finding two termed surd expressions?). Can anyone provide me with some pointers on what is going on here and p

.

\(\displaystyle \displaystyle{\sqrt{7 - 4\sqrt{3}}}\)

\(\displaystyle \displaystyle{\sqrt{4 +3 - 2*2*\sqrt{3}}}\)

\(\displaystyle \displaystyle{\sqrt{2^2 +(\sqrt{3})^2 - 2*2*\sqrt{3}}}\) → in the form of (a-b)² = a² + b² - 2ab

 

[Steven G]

Hi all,

I think this may just scrape into the intermediate algebra category.

I am looking at modulus questions and have found an example which I do not understand. The example is as follows:

"Find a simple surd expression for the positive square root of 7-4*sqrt[3].
7-4*sqrt[3] = (2 - sqrt[3])^2 or equivalently (sqrt[3]-2)^2. However, sqrt[ 7-4*sqrt[3]] = 2 - sqrt[3] not sqrt[3]-2 since by definition it must be positive."

The bit i do not understand is how they computed that 7-4*sqrt[3] = (2 - sqrt[3])^2. the rest makes sense to me.

I can see that both (2-sqrt[3])^2 and (sqrt[3]-2) equal 7-4*sqrt[3], however no matter how i manipulate it, I just can't see how they could have reasoned that it did.

I figure that there must be some method of manipulating surd expressions of which I am unaware. I know basic algebra with regards to radical expressions.

I don't even really know what to look up to find out what this type of thing is (finding two termed surd expressions?). Can anyone provide me with some pointers on what is going on here and p

Just note that (a-sqrt(b))^2 = a^2 + b - 2asqrt(b)

So (a-sqrt(b))^2 = a^2 + b -2asqrt(b) = 7-4sqrt(3)

Just solve 2a = 4 and b=3. and make sure that the non sqrt number is in the form a^2 + b

Then get a=2 and b=3

That is (2-sqrt(3))^2 = 7-4sqrt(3)

Let's make up one. Suppose we want to have 8sqrt(5). Then a=8/2=4 and b=5. Now a^2+b=21

Then 21-8sqrt(5) = (4-sqrt(5))^2.

Now it is simple.