Question

If $a \& b$ are rational numbers satisfying
$a + b\sqrt{75} = 7\bigl(\sqrt{108}-3\bigr)+\sqrt{27},$
then value of $a \& b$ respectively, are

• A. a=7, b=9
• B. a=45, b=-21
• C. a=-21, b=9
• D. a=21, b=9

Answer 1

Answer: (C)

$a=-21,\ b=9$

Answer 2

Step 1: Simplify the RHS

$$7\bigl(\sqrt{108}-3\bigr)+\sqrt{27} =7\bigl(6\sqrt{3}-3\bigr)+3\sqrt{3} =42\sqrt{3}-21+3\sqrt{3} =45\sqrt{3}-21.$$

Step 2: Express LHS in terms of $\sqrt{3}$

$$\sqrt{75}=5\sqrt{3},$$

so

$$a + b\sqrt{75}=a +5b\sqrt{3}.$$

Step 3: Equate rational and irrational parts
From

$$a +5b\sqrt{3}= -21 +45\sqrt{3},$$

we get the system:

• Rational part: $a=-21$
• Irrational part: $5b=45\implies b=9$.

Hence, $\boxed{a=-21,\ b=9}.$