1.4 - Surds and rationalising denominators

1.4 - Surds and rationalising denominators

Surds keep roots exact, so you can calculate without losing information to rounding. You will simplify, combine and multiply surds, then remove roots from denominators. You will also learn how to check that a different-looking answer still has the same value.

Exact roots and surds (Higher)

How can 45−1\frac{4}{\sqrt{5}-1} be exactly equal to 5+1\sqrt{5}+1? One looks like a fraction; the other does not. Keep this puzzle in mind as you build the tools to prove the equality.

You already know

A square root reverses squaring: 72=497^2=49, so 49=7\sqrt{49}=7. Expanding brackets means multiplying every term in one bracket by every term in the other.

A calculator gives 18≈4.243\sqrt{18}\approx4.243, but the rounded decimal is only close to the root. The symbol 18\sqrt{18} keeps its full value. Use == between exact equal values and ≈\approx before an approximation.

Surd

A surd is an irrational root written in exact form. An irrational number cannot be expressed as a fraction of integers with a nonzero denominator. For example, 18\sqrt{18} is a surd; 49=7\sqrt{49}=7 is rational and is not a surd.

For a positive integer, its square root is rational exactly when the integer is a perfect square. A square-root sign selects the nonnegative root. Thus 9=3\sqrt{9}=3, whereas solving x2=9x^2=9 gives x=3x=3 or x=−3x=-3. In real-number calculations, a\sqrt{a} requires a≥0a\geq0.

Here is a useful surprise: two irrational numbers can have a rational product. Both factors in 2×2=2\sqrt{2}\times\sqrt{2}=2 are irrational. “Contains a surd” does not mean “every result stays irrational”. These root facts are not supplied on the Higher formulae sheet; surds are not Foundation content.

Key idea

An exact root keeps all the information. A terminating decimal such as 4.2434.243 is rational and cannot be exactly equal to an irrational root.

You can test an alleged exact square root by squaring it. For example, 4.2432=18.0030494.243^2=18.003049, so it is close to 18\sqrt{18}, but is not equal to it.

Simplifying and combining surds (Higher)

A large root often hides a small root multiplied by a whole number. For example, 72=36×272=36\times2: its square factor 3636 is the part you can take out of the root.

Product of square roots

For a≥0a\geq0 and b≥0b\geq0,

ab=a×b\sqrt{ab}=\sqrt{a}\times\sqrt{b}

Both sides are nonnegative and square to abab, which explains the equality. Higher: not supplied. Not in the formulae sheet: learn it.

Choose the largest square factor if you can. A fully simplified integer surd has no square factor greater than 11 left inside the root. Then combine like surds, which have the same root part. Just as 6x+3x=9x6x+3x=9x, 62+32=926\sqrt{2}+3\sqrt{2}=9\sqrt{2}.

Worked example 1: make the roots alike

Question. Simplify 72+18\sqrt{72}+\sqrt{18}.

Plan. Extract a square factor from each root, then add their coefficients.

72+18=36×2+9×2find square factors=62+32take their roots=92add coefficients\begin{aligned} \sqrt{72}+\sqrt{18} &=\sqrt{36\times2}+\sqrt{9\times2} && \text{find square factors}\\ &=6\sqrt{2}+3\sqrt{2} && \text{take their roots}\\ &=9\sqrt{2} && \text{add coefficients} \end{aligned}

Why simplify first? It reveals that both terms are multiples of the same number, 2\sqrt{2}.

Answer. 929\sqrt{2}.

Check. 72≈8.485\sqrt{72}\approx8.485 and 18≈4.243\sqrt{18}\approx4.243; their sum is about 12.72812.728, as is 929\sqrt{2}.

Now change only the operation from addition to subtraction. The square factors stay the same; the coefficients must be subtracted instead.

Worked example 2: subtract like surds

Question. Simplify 72−18\sqrt{72}-\sqrt{18}.

Plan. Simplify each root before subtracting.

72−18=62−32extract square factors=(6−3)2keep the root part=32subtract coefficients\begin{aligned} \sqrt{72}-\sqrt{18} &=6\sqrt{2}-3\sqrt{2} && \text{extract square factors}\\ &=(6-3)\sqrt{2} && \text{keep the root part}\\ &=3\sqrt{2} && \text{subtract coefficients} \end{aligned}

Why keep the root part? Subtraction changes how many copies you have, not the value of each copy.

Answer. 323\sqrt{2}.

Check. Adding 323\sqrt{2} back to 18=32\sqrt{18}=3\sqrt{2} gives 62=726\sqrt{2}=\sqrt{72}.

Multiplication has a root rule; addition has no matching rule. This distinction is the main trap here.

Watch out

The line 9+16=9+16\sqrt{9+16}=\sqrt{9}+\sqrt{16} is wrong: the left side is 55, while the right side is 77. Take the root of the whole sum, or simplify separate root terms without joining their radicands.

In your turn, keep the subtraction method and change the radicands. Show the simplified roots so your method is visible.

Products and surd brackets (Higher)

When you multiply surds, multiply outside coefficients separately from the roots. For example, 6×15=90=310\sqrt{6}\times\sqrt{15}=\sqrt{90}=3\sqrt{10}. In brackets, the same distributive law you use with algebra still applies.

Conjugates

Expressions such as u+vu+v and u−vu-v are conjugates: the same two terms, with the sign between them reversed. For example, 5+2\sqrt{5}+2 and 5−2\sqrt{5}-2 are conjugates; 5+2\sqrt{5}+2 and 5−3\sqrt{5}-3 are not.

Expanding conjugates creates opposite cross terms, which cancel. Expanding a square creates equal cross terms, which add. You will need both patterns.

Two bracket identities

(u+v)(u−v)=u2−v2(u+v)2=u2+2uv+v2\begin{aligned} (u+v)(u-v)&=u^2-v^2\\ (u+v)^2&=u^2+2uv+v^2 \end{aligned}

These hold for real u,vu,v. Higher: not supplied. Not in the formulae sheet: learn it.

Start with two different brackets. Keeping all four products visible makes the signs easier to control.

Worked example 3: four products

Question. Expand and simplify (5+2)(5−3)(\sqrt{5}+2)(\sqrt{5}-3).

Plan. Multiply each term in the first bracket by each term in the second, then collect like terms.

(5+2)(5−3)=5−35+25−6four products=(5−6)+(−3+2)5group like terms=−1−5simplify\begin{aligned} (\sqrt{5}+2)(\sqrt{5}-3) &=5-3\sqrt{5}+2\sqrt{5}-6 && \text{four products}\\ &=(5-6)+(-3+2)\sqrt{5} && \text{group like terms}\\ &=-1-\sqrt{5} && \text{simplify} \end{aligned}

Why is the first product 55? Multiplying 5\sqrt{5} by itself reverses the root.

Answer. −1−5-1-\sqrt{5}.

Check. The first bracket is positive and the second negative, so the product must be negative. The answer has that sign.

Change only the constant in the second bracket, from −3-3 to 22. Now the brackets match: you are squaring a sum.

Worked example 4: a squared surd bracket

Question. Expand and simplify (5+2)2(\sqrt{5}+2)^2.

Plan. Write two identical brackets and include both cross products.

(5+2)2=5+25+25+4four products=5+4+45collect terms=9+45add integers\begin{aligned} (\sqrt{5}+2)^2 &=5+2\sqrt{5}+2\sqrt{5}+4 && \text{four products}\\ &=5+4+4\sqrt{5} && \text{collect terms}\\ &=9+4\sqrt{5} && \text{add integers} \end{aligned}

Why is there a middle term? Each bracket supplies one cross product; together they give 454\sqrt{5}.

Answer. 9+459+4\sqrt{5}.

Check. 5>2\sqrt{5}>2, so the bracket exceeds 44 and its square exceeds 1616. The answer is about 17.94417.944, which fits.

A quick sign or size check helps, but it cannot replace the requested algebra.

Watch out

(5+2)2=5+4(\sqrt{5}+2)^2=5+4 misses both cross products. Squaring a sum is multiplying two whole brackets. The correct expansion is 9+459+4\sqrt{5}.

Try the same squared-bracket structure with a different root.

Rationalising denominators (Higher)

You can change a denominator without changing a fraction's value. Multiply the numerator and denominator by the same nonzero expression: their extra factors divide to give 11.

Rationalising the denominator

Rationalising rewrites a fraction with a rational denominator. It does not promise that the whole value is rational. For example, 12=22\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} still has an irrational value.

For a single root denominator, multiply by that root. For a two-term surd denominator, use its conjugate. The difference-of-squares identity removes the remaining root from the denominator. Check that the original denominator and the multiplying expression are nonzero. These methods are not supplied on the Higher formulae sheet.

Worked example 5: a single-root denominator

Question. Rationalise the denominator of 45\frac{4}{\sqrt{5}}.

Plan. Multiply top and bottom by 5\sqrt{5}, whose square is rational.

45=455×5multiply by one=455square the root\begin{aligned} \frac{4}{\sqrt{5}} &=\frac{4\sqrt{5}}{\sqrt{5}\times\sqrt{5}} && \text{multiply by one}\\ &=\frac{4\sqrt{5}}{5} && \text{square the root} \end{aligned}

Why change the numerator too? Otherwise you would change the fraction's value.

Answer. 455\frac{4\sqrt{5}}{5}.

Check. Multiplying this answer by the original denominator gives 4×55=4\frac{4\times5}{5}=4, the original numerator.

Now change only the denominator by subtracting 11. Multiplying by 5\sqrt{5} alone would leave a root below the line, so choose the conjugate instead.

Worked example 6: a two-term denominator

Question. Rationalise the denominator of 45−1\frac{4}{\sqrt{5}-1}.

Plan. Multiply top and bottom by 5+1\sqrt{5}+1, then use a difference of squares.

45−1=4(5+1)(5−1)(5+1)use the conjugate=4(5+1)5−1cross terms cancel=5+1cancel the factor four\begin{aligned} \frac{4}{\sqrt{5}-1} &=\frac{4(\sqrt{5}+1)}{(\sqrt{5}-1)(\sqrt{5}+1)} && \text{use the conjugate}\\ &=\frac{4(\sqrt{5}+1)}{5-1} && \text{cross terms cancel}\\ &=\sqrt{5}+1 && \text{cancel the factor four} \end{aligned}

Why use opposite signs? The two cross products cancel, leaving the rational denominator 44.

Answer. 5+1\sqrt{5}+1. This proves the equality in the opening puzzle.

Check. (5+1)(5−1)=5−1=4(\sqrt{5}+1)(\sqrt{5}-1)=5-1=4. Multiplying back recovers the numerator exactly.

A rational denominator is the target. An exact root may still remain in the numerator.

Watch out

The line (5−1)2=5−1(\sqrt{5}-1)^2=5-1 is wrong: equal signs in the brackets make cross terms add. The conjugate product is (5−1)(5+1)=4(\sqrt{5}-1)(\sqrt{5}+1)=4.

For Pearson questions asking you to show working, write the multiplying fraction and the denominator expansion. A calculator's exact-looking final answer alone does not demonstrate rationalisation.

Exam tip

If the question specifies a form such as a+bca+b\sqrt{c}, simplify into that form after rationalising. Keep exact roots; a rounded decimal does not meet an exact-form instruction.

Change the numerator for your own rationalisation. That changes the final coefficient but leaves the conjugate decision intact.

Quick Recap

  • A surd is an irrational root kept exact; a square-root sign chooses the nonnegative root.
  • Extract square factors, then combine only like root parts.
  • Expand every bracket product, including cross terms.
  • Rationalise by multiplying top and bottom by the same nonzero factor; use a conjugate for a two-term denominator.

The opening fraction equals 5+1\sqrt{5}+1 because conjugate multiplication makes its denominator 5−1=45-1=4, which cancels the numerator's factor 44.