Simplify √45 − √12 − √10/√2 + 3/√3 into the form a√3 + b√5 with a and b rational, and find a + b.

Step-by-step solution

1. 1

   Simplify the two plain surds. Pull out the largest square factor:

   $\sqrt{45}=\sqrt{9\cdot 5}=3\sqrt5,\qquad \sqrt{12}=\sqrt{4\cdot 3}=2\sqrt3$

2. 2

   Combine the quotient of surds into one radical. Using $\dfrac{\sqrt a}{\sqrt b}=\sqrt{\dfrac ab}$:

   $\frac{\sqrt{10}}{\sqrt2}=\sqrt{\frac{10}{2}}=\sqrt5$

   This is faster than rationalising the denominator first.

3. 3

   Rationalise the last term.

   $\frac{3}{\sqrt3}=\frac{3\sqrt3}{3}=\sqrt3$

   (Equivalently $\tfrac{3}{\sqrt3}=\sqrt3$ because $\sqrt3\cdot\sqrt3=3$.)

4. 4

   Collect the √3 terms and the √5 terms separately. They are unlike surds and cannot be merged:

   $3\sqrt5-2\sqrt3-\sqrt5+\sqrt3=(3-1)\sqrt5+(-2+1)\sqrt3=2\sqrt5-\sqrt3$

5. 5

   Match the required form and add. Writing it as $a\sqrt3+b\sqrt5$ gives $a=-1$ and $b=2$, so

   $a+b=-1+2=1$

   $\boxed{a+b=1}$

6. 6

   Check numerically. $\sqrt{45}=6.7082$, $\sqrt{12}=3.4641$, $\tfrac{\sqrt{10}}{\sqrt2}=2.2361$, $\tfrac{3}{\sqrt3}=1.7321$. The sum is $6.7082-3.4641-2.2361+1.7321=2.7401$, and $2\sqrt5-\sqrt3=4.4721-1.7321=2.7401$ ✓.