Can someone explain surds to me and also show me using this example: 3square root15 divided by square root 3?

Answer 1

A surd is basically a number which is left as "square root of x" (which from now on I will call sqrt x). So sqrt2 is a surd, whereas the decimal answer to that (1.414.....) isn't.

A proper surd, I believe has the sqrt on the top of a fraction. If you have one on the bottom of a fraction, you have to get rid of it! How?, I hear you ask. Well, if you take y/y then that is always equal to 1. If you multiply a fraction by 1, it stays the same. Therefore if you have sqrt x on the bottom, you multiply by sqrtx/sqrtx. (NB Do not cancel sqrtx/sqrtx=1, that will not help).

So in your example, we will multiply by sqrt3. 3*sqrt15/sqrt3=3*sqrt15/sqrt3 * sqrt3/sqrt3
So the top of the fraction becomes 3*sqrt15*sqrt3, and the bottom becomes sqrt3*sqrt3.

We'll start by simplifying the bottom. sqrt3*sqrt3=3 (thats the definition of a square root), so the bottom of the equation is3.

For the top of the equation you can use the relationship a*b=sqrt(a^2*b^2). Therefore the top becomes 3*sqrt15*sqrt3=3*sqrt(sqrt15^2*sqrt3^2). We know that (sqrtx)^2=x, so the top becomes 3*sqrt (15*3)=3*sqrt(45).

So the whole fraction becomes 3*sqrt45/3=sqrt45

Which sounds quite good, but we're not quite there yet.

Hint for more difficult questions, which are sure to follow:
sqrt(a^2*b)=a*sqrtb (which uses one of the rules I told you earlier).
So sqrt(45)=sqrt(9*5)=sqrt(3^2*5)=3*sqrt5, which is the final answer.

Ta da!

Answer 2

The way I would attempt your question is to square each term, then do the arithmetic and then take the square root.

So 3*sqrt(15)/sqrt(3) squared becomes 9*15/3 which equals 45. Then taking the square root gives sqrt(45).

This can be simplified a bit more if you like
sqrt(45) = sqrt(9*5) = sqrt(9)*sqrt(5) = 3*sqrt(5).

If the denominator is of the form a+sqrt(b) then to get rid of the sqrt(b) in the denominator we use what is called the conjugate. The conjugate of a+sqrt(b) is a-sqrt(b), notice all that has happened is the sign of sqrt(b) has changed.

To use the conjugate we would multiply the given surd by a-sqrt(b)/a-sqrt(b). Notice how we are actually multiplying by 1, so this doesn't change the value of the surd, but it gets rid of the sqrt(b) in the denominator.

The conjugate is also used when dealing with division of complex numbers.

Answer 3

3square root15 divided by square root 3
=3*sqrt15/sqrt3
=3*sqrt3*sqrt5/sqrt3
=3*sqrt5
=3*5^(1/2)
=6.71
A surd is a number which is said as "square root of x" (sqrt x). So sqrt2 is a surd, whereas the decimal answer to that (1.4141) is not be a surd.

Answer 4

Surds deal with irrational numbers, ie ones which you go on calculating.
3 sqrt 15 / sqrt 3. Multiply the denominator by sqrt 3, as we want to express the number in simplest form. cancel common terms.We get sqrt 45, ie 3 sqrt 5. Now that looks nicer!

Answer 5

let me try something simple.

square root 15 is 15^(1/2). now you can split terms up. that is how surds work. so 15^(1/2) = 3^(1/2) * 5^(1/2)

thus ( 3 square root 15) /(square root 3) = 3 * 3^(1/2) * 5^(1/2) / 3^(1/2)

cancelling 3^(1/2), you get = 3 * 5^(1/2)

Answer 6

I hope this helps:

sqrt(3) = 3^(1/2);
1/sqrt(3) = 3^(-1/2);

so:

3·sqrt(15)/sqrt(3) =
3·sqrt(5·3)·3^(-1/2) =
3·(5·3)^(1/2)·3^(-1/2) =
3·5^(1/2)·3^(1/2)·3^(-1/2) =
3·5^(1/2) =
3·sqrt(5)

:)

Answer 7

sqrt(15) equals sqrt(5) times sqrt(3).
Therefore, 3*sqrt(15) equals 3* sqrt(5)*sqrt(3)
Divide by sqrt(3) and you get:
3*sqrt(5).

Answer 8

3*15^(1/2)/3^(1/2)=3*3^(1/2)*5^(1/2)/3^(1/2)=3*5^(1/2)