Let a=4+√15. Find the value of a^3-1/a^3.

Let $a = 4 + 15$ . Find the value of $a^{3} - \frac{1}{a ^{3}}$ .

hard 3 marks Class 9 · Mathematics · Number Systems

Answer

12615

The correct answer is $12615$ — Multiply $215 \times 63$ to get $12615$ .

Find the reciprocal: $\frac{1}{4 + 15} = \frac{4 - 15}{( 4 + 15 ) ( 4 - 15 )} = \frac{4 - 15}{16 - 15} = 4 - 15$ .
Note that $1/ a$ equals the conjugate of $a$ : $1/ a = 4 - 15$ .
Compute $a - 1/ a = (4 + 15) - (4 - 15) = 215$ .
Compute $a^{2} = (4 + 15)^{2} = 31 + 815$ .
Compute $(1/ a)^{2} = (4 - 15)^{2} = 31 - 815$ .
Compute $a^{2} + (1/ a)^{2} + a (1/ a) = (31 + 815) + (31 - 815) + 1 = 63$ .
Use identity: $(a - 1/ a) (a^{2} + (1/ a)^{2} + a (1/ a)) = a^{3} - 1/ a^{3}$ .
Multiply $215 \times 63$ to get $12615$ .

Common mistake: Assuming the reciprocal equals the original number

Concept tested: Algebraic manipulation with conjugates