Question

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Help?

Answer 1

a square root of a perfect square number always reduces to a rational number. Otehrwise we call these surds which are irrational.

Answer 2

you may need to multiply surds together as well to see more clearly

Generally
\[\sqrt a \times \sqrt a = (\sqrt a )^2 = a\]
So eg. \[\sqrt 7 \times \sqrt 7 = \sqrt 49 = 7\] which is rational where the original (surds) were not rational

Answer 3

Okay

Answer 4

May you explain more?

Answer 5

well let's work through them one by one.
First one, what is the answer?

Answer 6

Sqrt 5 x sqrt 5
so that's 25, right?

Answer 7

sqrt 25 precisely. what does this simplify to?

Answer 8

5

Answer 9

5 is rational, right?

Answer 10

great. ok actually I think you should first understand what a rational number is

Answer 11

IF a number is rational, it just means you can write the number as a fraction
where both numerator and denominator are intgers (whole numbers) and the denominator is non-zero.

All terminating and recurring decimals, integers are rational.
Irrational numbers are non-terminating, non recurring, things like pi, and sqrts of non-perfect square numbers eg. sqrt(10)

Answer 12

Okay, I got it!

Answer 13

so your job is to see whether your sqrt is of a perfect square or not and can do this by eye or by multiplying them out into one sqrt

Remember the perfect square numbers are 1, 4, 9, 16, 25 etc.
when you take their sqrt, it is a whole number.

Answer 14

The next one is sqrt48 and there is no perfect square so it's irrational

Answer 15

excellent!

Answer 16

The last two I do not get :/
I forgot how to do it with that number in front :/

Answer 17

With the number at the front, you can rewrite that as sqrt(a^2) eg. 3 is the same as sqrt(9)

However don't do it like this, ignore the whole number for now and focus on the surds.

Answer 18

So it's 7sqrt9

Answer 19

simplify is further. that is just 7 times 3 isn't it? so 21

Answer 20

yeah
now the sqrt of 21, right?

Answer 21

nope jut 21 because we already treated the sqrt(9) as a 3. so we no longer need the sqrt

Answer 22

so 21 is rational so c is another choice?

Answer 23

exactly

Answer 24

once you finish the last one, I'll teach you a neat shortcut to these kinds

Answer 25

9 x 4 is sqrt36 is 6 so 2 x 6 is 12 so it's rational

Answer 26

good job!
Ok so the shortcut is simple:

rational x rational = rational
irrational x rational = irrational

irrational x irrational = rational but ONLY if either:
1) both are the same surd eg. sqrt(3) x sqrt(3)
2) the binomial product is in conjugate, becomes a difference of two squares:
\[(a+\sqrt b ) (a- \sqrt b) = a^2 - (\sqrt b)^2 = a^2 - b\]

You'll learn about 2 later.

Answer 27

you can see for yourself that the rule above works, so you don't even need to multiply them out individually and get the final answer to tell if it is rational or not. That's why I said you need not worry about the numbers in front.