# Help with surds maths question?

π is an irrational number and so is √3. Therefore, determine whether

(π − √3)(π + √3) is an irrational number.

### 9 Answers

- lenpol7Lv 75 months ago
(pi - sqrt(3)(pi _ sqrt(3)

Multiply together using FOIL

pi^2 - pisqrt(30 + pisqrt(3) - 3

Hence collecting terms

pi^2 - 3

9.869604401.... - 3

= 6.869604401.... which is irrational!!!!

- IndikosLv 75 months ago
pi^2 is irrational

see https://planetmath.org/piandpi2areirrational

therefore (π − √3)(π + √3) = π^2 - 3 is irrational

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- rotchmLv 75 months ago
Developing as others have here for you, you get the answer π² - 3.

You must now verify if this is irrational or not.

If it were rational then π² - 3 = a/b for some integers a & b. Thus

π² = a/b + 3 = rational = A/B for integers A & B.

Thus you question becomes one in finding if π² is rational or not.

For this, depends on what tools you have (what grade you are in). Just from your info, we cant

say if π² is rational or not. [But yes, π² is irrational. This can be shown by various means. Perhaps the simplest relies on knowing that π is transcendental].

- LônLv 75 months ago
(π − √3)(π + √3)

π(π − √3) + √3 (π - √3)

π^2 - √3π + √3π - 3

π^2 - 3

So it is an irrational number as you can't express it as a fraction.