Additional Mathematics Synthesis Guide 29: Surds, Conjugates, Rationalising Denominators, Surd Equations and Exact Verification

Simplify → preserve exactness → use conjugates where useful → solve with restrictions → verify in the original equation.

1. What a surd is

A surd is an irrational radical expression that is kept in exact form. Examples include √2, 3√5 and 2+√7. The decimal 1.414… may approximate √2, but √2 is the exact value.

Exact forms are valuable because identities remain visible. For example, (√2)²=2 exactly, while squaring a rounded decimal may introduce approximation error.

2. Simplify radicals before combining

For √72, factor out the largest convenient square:

√72=√(36·2)=6√2.

Similarly, √50=5√2 and √8=2√2, so

√50−√8=3√2.

Unlike radicals cannot be combined: √2+√3 does not simplify to √5.

3. Multiplication follows radical structure

For non-negative real quantities,

√a·√b=√(ab).

Thus

(3√2)(2√6)=6√12=12√3.

Simplify at the end or during the calculation, but preserve exact structure.

4. Conjugates

The conjugate of a+b√c is a−b√c. Their product is

(a+b√c)(a−b√c)=a²−b²c.

The surd terms cancel. This is why conjugates are powerful for rationalising binomial denominators.

5. Worked conjugate product

Evaluate

(5+2√3)(5−2√3).

Difference of squares gives

25−12=13.

The result is rational even though each factor is irrational.

6. Rationalising a monomial denominator

For

3/√5,

multiply numerator and denominator by √5:

3√5/5.

The value is unchanged because we multiplied by √5/√5=1.

7. Rationalising a binomial denominator

Simplify

1/(2+√3).

Multiply by the conjugate:

(2−√3)/(2−√3).

The denominator becomes 4−3=1, so

1/(2+√3)=2−√3.

A numerical check gives both sides approximately 0.268, supporting the exact identity.

8. Worked rationalisation with coefficients

Simplify

4/(3−√5).

Multiply by 3+√5:

4(3+√5)/(9−5)=3+√5.

Recombining confirms (3+√5)(3−√5)=4.

9. Comparing surd expressions by exact algebra

To compare 3√2 and 2√5, both are positive, so squaring preserves order:

(3√2)²=18, (2√5)²=20.

Therefore 3√2<2√5.

The positivity condition matters. Squaring is order-preserving only when the signs are controlled appropriately.

10. Coefficient comparison in a surd basis

If a+b√2=c+d√2 with rational a,b,c,d, then equality implies a=c and b=d.

Otherwise we would obtain a non-zero rational multiple of √2 equal to a rational number, contradicting irrationality.

This supports parameter problems where two exact surd forms are equated.

11. Worked coefficient comparison

If

(a+b√3)(2−√3)=7−4√3,

expand:

(2a−3b)+(2b−a)√3=7−4√3.

Hence

• 2a−3b=7
• 2b−a=−4

Solving gives b=1 and a=6.

12. Surd equations begin with domain control

For

√(x+1)=x−1,

the left side requires x≥−1, but the right side must also be non-negative because it equals a square root. Therefore x≥1.

That stronger condition should be recorded before squaring.

13. Squaring can create extraneous roots

For √(x+1)=x−1, square both sides:

x+1=x²−2x+1.

So x²−3x=0, giving x=0 or 3.

But x=0 violates x≥1 and fails the original equation. Only

x=3

survives.

14. Why squaring is not fully reversible

If A=B, then A²=B². But A²=B² allows A=B or A=−B. Squaring removes sign information.

Therefore a surd equation solved by squaring must be checked in the original unsquared equation.

15. Isolate one radical before squaring

For

√(x+4)+1=x,

first isolate:

√(x+4)=x−1.

Then record x≥1 before squaring. Isolating the radical makes the sign condition visible and keeps the algebra controlled.

16. Two radicals may require two stages

An equation such as

√(x+5)−√x=1

with x≥0 can be solved by isolating one radical, squaring, then isolating again if necessary.

At every squaring stage, the transformed equation may gain candidates. Final substitution into the original equation is essential.

17. Worked two-radical equation

Solve √(x+5)−√x=1.

Move √x:

√(x+5)=√x+1.

Square:

x+5=x+1+2√x.

Thus 4=2√x, so √x=2 and x=4.

Check: √9−√4=3−2=1. Therefore x=4.

18. Surd identities can be verified by squaring

Show that

√(7+4√3)=2+√3.

Both sides are positive. Square the proposed right side:

(2+√3)²=4+3+4√3=7+4√3.

Since the positive square root is unique, the identity holds.

19. Reverse engineering nested surds

To simplify √(a+b√c), try a form √m+√n. Squaring gives

m+n+2√(mn).

So seek m+n=a and 2√(mn)=b√c.

For 7+4√3, choose m=4,n=3.

20. Exact geometry often produces surds

A distance such as √13 is not an unfinished answer merely because it is irrational. If the question requests exact form, √13 is often preferable to 3.606.

Approximation should come only at the requested stage. Exact form preserves relationships for later algebra, trigonometry or calculus.

21. Common failure patterns

• Writing √a+√b=√(a+b).
• Combining unlike surds before simplification.
• Using the same sign rather than the conjugate sign when rationalising a binomial denominator.
• Approximating before exact algebra is complete.
• Squaring a surd equation without recording sign and domain conditions.
• Accepting every root of the squared equation.
• Comparing squared values when the original signs are not controlled.
• Forgetting to verify a nested-surd identity using the positive-root condition.

22. A reliable exact-surd routine

1. Simplify radicals by extracting square factors.
2. Combine only like surd terms.
3. Use conjugates for binomial denominator rationalisation.
4. Preserve exact form through algebra.
5. For equations, record radical domains and sign conditions before squaring.
6. Isolate radicals before applying non-reversible operations.
7. Check all candidates in the original equation.
8. Approximate only when requested or useful for a plausibility check.

23. Practice set

1. Simplify √98.
2. Simplify 3√8−√18.
3. Evaluate (4+√7)(4−√7).
4. Rationalise 5/√3.
5. Rationalise 1/(3+√2).
6. Rationalise 6/(2−√3).
7. Compare 4√3 and 3√5 exactly.
8. If a+b√2=7−3√2, find a,b.
9. Solve √(x+2)=x.
10. Solve √(x+5)=x−1.
11. Solve √(x+5)−√x=1.
12. Explain why squaring can create extraneous roots.
13. Show √(5+2√6)=√3+√2.
14. Find m,n if √(9+4√5)=√m+√n with positive m,n.
15. Why is √13 an exact answer?
16. Why does rationalising a denominator not change the value?
17. What is the conjugate of 5−2√7?
18. Why must both sides be positive when proving √A=B by squaring?
19. Give one numerical verification for 1/(2+√3)=2−√3.
20. Why should approximation be delayed?

Answers

1. 7√2.
2. 6√2−3√2=3√2.
3. 16−7=9.
4. 5√3/3.
5. (3−√2)/7.
6. 6(2+√3)/(4−3)=12+6√3.
7. Squares are 48 and 45; both positive, so 4√3>3√5.
8. a=7,b=−3.
9. Need x≥0. Squaring gives x+2=x², so x=2 or −1; only x=2 survives.
10. Need x≥1. Squaring gives x+5=x²−2x+1, so x²−3x−4=0; x=4 or −1, only x=4 survives.
11. x=4.
12. A²=B² allows A=±B, so sign information is lost.
13. (√3+√2)²=5+2√6, and the right side is positive.
14. m+n=9 and mn=20, so {m,n}={4,5}.
15. It names the exact positive number whose square is 13, without rounding.
16. We multiply by a form equal to 1, such as the conjugate divided by itself.
17. 5+2√7.
18. Squaring alone cannot distinguish B from −B; the positive square root condition selects the correct sign.
19. Both sides are approximately 0.268.
20. Exact structure supports cancellation, comparison and later algebra without accumulating rounding error.

24. What mastery looks like

Mastery means the learner can simplify and compare surds exactly, use conjugates because of difference-of-squares structure, preserve restrictions through radical equations, identify where squaring loses information and verify every candidate in the original equation.

The transfer test is to remove the instruction “rationalise” or “solve the surd equation”. If the learner independently chooses exact simplification, conjugates or controlled squaring because those representations make the target easier, surd algebra has become part of method selection.

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Continue through Batch 08

• Guide 30: Integration Rule Selection — Rational Powers, Exponentials, Trigonometric Forms and Reverse Chain Rule
• Guide 31: Increasing and Decreasing Functions, Derivative Sign Charts, Stationary Inflexions and Second-Derivative Decisions
• Guide 32: Transformed Trigonometric Graphs — Amplitude, Period, Midline, Tangent Structure and Modelling
• Additional Mathematics Directory
• BTT Mathematics Hub