Area of ∆ l^{e }= 1/2[x_{1}(y_{2} − y_{3} ) + x_{2} (y_{3} − y_{1}) + x_{3} (y_{1} − y_{2} )],
∆ ^{le} = 1/2 [(20 + 4) − (−4 − 3) + 2(3 − 0)],
=1/2(8 + 7 + 6) = 21/2 sq units.
We have, 5^{x − 3} × 3^{2x − 8} = 225,
⇒ 5^{x − 3} × 3^{2x − 8 }= 5^{2} × 3^{2}
⇒ x – 3 = 2 and 2x – 8 = 2 [on equating the exponents],
⇒x=5.
Let x = 5 √2 be a rational number,
x / 5 = √2,
x= 3.222.
Let us consider two irrational numbers 3 + √2 and 3 - √2
(3 + √2 ) + (3 − √2) = 3 + √2 + √2 = 6, which is a rational number.
We know that, if a and b are two distinct positive irrational numbers,
Then √ab is an irrational number lying between a and b.
∴ Irrational number between √2 and √3 is √(√2 x √3) = √√6 = 6^{1/4}
Irrational number between √2 and 6^{1/4} is √(√2 x 6^{1/4}) = 2^{1/4} x 6^{1/8}
Hence, required irrational numbers are 6^{1/4} and 2^{1/4} × 6^{1/8}.
No, Let us consider a rational number x = 0 and an irrational number
y = √2 then,
xy = 0 x √2, which is not irrational.