Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — simplifying and combining surds
Simplify 50−18+8.
- 150=25×2=52Look for the largest square factor. 25 is a square, 50 is not.
- 218=9×2=32Same idea with 9.
- 38=4×2=22All three reduce to multiples of 2, which is what makes them combinable.
- 452−32+22=42Treat 2 as a like term and just handle the coefficients.
Example 2 — rationalising with a conjugate
Rationalise the denominator of 5−26.
- 1The conjugate of 5−2 is 5+2.Same two terms, middle sign reversed.
- 25−26×5+25+2Multiplying by 1, so the value is unchanged — but you must do the top as well as the bottom.
- 3Denominator: (5)2−22=5−4=1The conjugate turns it into a difference of two squares, and squaring removes the root.
- 4=6(5+2)=65+12With a denominator of 1 there is nothing left to divide by.
Example 3 — a logarithmic equation, including the check
Solve log2x+log2(x−2)=3.
- 1log2[x(x−2)]=3A sum of logs is the log of the product.
- 2x(x−2)=23=8Rewrite in index form to remove the logarithm.
- 3x2−2x−8=0⇒(x−4)(x+2)=0An ordinary quadratic now.
- 4x=4 or x=−2Two candidates — but only candidates until they are checked.
- 5x=−2 would need log2(−2), which does not exist. So x=4.Check: log24+log22=2+1=3. The checking step is compulsory.