Rationalising Two-Term Denominators

Use the same two terms with the sign between them reversed.

1. The fraction \dfrac{3-\sqrt2}{3-\sqrt2} equals 1, so it keeps the value unchanged.
2. The new denominator is (3+\sqrt2)(3-\sqrt2)=9-2=7, which is rational.

Multiply top and bottom by the expression with the sign reversed.

1. Multiply the numerator and denominator by 2-\sqrt3, the denominator with its sign reversed.
2. The denominator is a difference of two squares: (2+\sqrt3)(2-\sqrt3)=2^2-(\sqrt3)^2=4-3=1.
3. The numerator is 1\times(2-\sqrt3), so the answer is \frac{2-\sqrt3}{1}=2-\sqrt3.

Use the difference of two squares to remove the surd from the denominator.

1. Multiply top and bottom by 3+\sqrt5.
2. The denominator becomes 9-5=4, giving \dfrac{4(3+\sqrt5)}4.
3. Cancel the factor 4 to get 3+\sqrt5.

Change the plus sign to a minus sign in the rationalising factor.

1. Multiply top and bottom by \sqrt7-1.
2. The denominator is 7-1=6, so the fraction is \dfrac{2(\sqrt7-1)}6.
3. Cancel the factor 2 to get \dfrac{\sqrt7-1}3.

Multiply top and bottom by the sum of the two denominator terms.

1. Multiply top and bottom by \sqrt6+1.
2. The denominator is 6-1=5.
3. Cancelling the factor 5 leaves \sqrt6+1.

Keep track of the sign when you subtract the square of the surd.

1. Multiply top and bottom by 2-\sqrt7.
2. The denominator is 4-7=-3, giving \dfrac{3(2-\sqrt7)}{-3}.
3. Dividing by -3 gives \sqrt7-2.

Square the whole surd term, including its coefficient, in the denominator.

1. Multiply top and bottom by 5-2\sqrt3.
2. The denominator is 25-(2\sqrt3)^2=25-12=13.
3. Expand the numerator to get \dfrac{30-12\sqrt3}{13}.

Multiply the surd numerator by both terms of the rationalising factor.

1. Multiply the numerator and denominator by \sqrt5-2.
2. Numerator: \sqrt5(\sqrt5-2)=5-2\sqrt5.
3. Denominator: (\sqrt5+2)(\sqrt5-2)=5-4=1.
4. So the answer is 5-2\sqrt5.

The rationalising factor is the same as the numerator, so expand a square carefully.

1. Multiply top and bottom by \sqrt3+1.
2. The numerator is (\sqrt3+1)^2=4+2\sqrt3, and the denominator is 3-1=2.
3. Divide both numerator terms by 2 to get 2+\sqrt3.

Simplify and collect like surds in the denominator first.

1. Since \sqrt8=2\sqrt2, the denominator simplifies to \sqrt2.
2. Rationalise \dfrac1{\sqrt2} by multiplying top and bottom by \sqrt2.
3. The answer is \dfrac{\sqrt2}{2}.

Divide by the coefficient of the unknown, then rationalise.

1. Divide by 4+\sqrt7 to get x=\dfrac3{4+\sqrt7}.
2. Multiply top and bottom by 4-\sqrt7.
3. The denominator becomes 16-7=9, so x=\dfrac{3(4-\sqrt7)}9=\dfrac{4-\sqrt7}3.

Divide the area by the length and rationalise the resulting fraction.

1. The width is \dfrac{10}{\sqrt{11}+1} cm.
2. Multiply top and bottom by \sqrt{11}-1.
3. The denominator is 11-1=10, so the width is \sqrt{11}-1 cm.

The same difference-of-squares method works when both terms are surds.

1. Multiply top and bottom by \sqrt6-\sqrt2.
2. The denominator is 6-2=4.
3. Cancel the factor 2 in \dfrac{2(\sqrt6-\sqrt2)}4 to get \dfrac{\sqrt6-\sqrt2}2.

Multiply both sides by the denominator and expand the product.

1. Multiply both sides by 3+\sqrt2: k=(6-2\sqrt2)(3+\sqrt2).
2. Expand: 6\times3=18, 6\times\sqrt2=6\sqrt2, -2\sqrt2\times3=-6\sqrt2 and -2\sqrt2\times\sqrt2=-4.
3. The surd terms cancel, so k=18-4=14.

Rationalise both fractions before adding them.

1. For the first fraction, multiply top and bottom by \sqrt5+2; its denominator becomes 5-4=1, leaving \sqrt5+2.
2. The second fraction similarly becomes \sqrt5-2.
3. Adding gives 2\sqrt5.

Expand the entire numerator after multiplying by the conjugate.

1. Multiply top and bottom by \sqrt3-1.
2. The numerator expands to 6-2\sqrt3+5\sqrt3-5=1+3\sqrt3.
3. The denominator is 3-1=2, so the answer is \dfrac{1+3\sqrt3}2.

Multiply by the conjugate, then simplify the products of unlike surds.

1. Multiply top and bottom by 2\sqrt5+\sqrt3.
2. The denominator is (2\sqrt5)^2-(\sqrt3)^2=20-3=17.
3. The numerator expands to 2\sqrt{75}+\sqrt{45}-2\sqrt5-\sqrt3.
4. Simplify \sqrt{75}=5\sqrt3 and \sqrt{45}=3\sqrt5, then collect terms to get \dfrac{9\sqrt3+\sqrt5}{17}.

Rationalise the first denominator, then multiply and collect terms.

1. Rationalise the first fraction: multiply the numerator and denominator by \sqrt6+\sqrt2. The denominator becomes 6-2=4, so \frac{1}{\sqrt6-\sqrt2}=\frac{\sqrt6+\sqrt2}{4}.
2. Multiply by \frac{\sqrt3}{\sqrt2}: \frac{(\sqrt6+\sqrt2)\sqrt3}{4\sqrt2}=\frac{\sqrt{18}+\sqrt6}{4\sqrt2}.
3. Divide each term of the numerator by \sqrt2: \frac{\sqrt{18}}{\sqrt2}=\sqrt9=3 and \frac{\sqrt6}{\sqrt2}=\sqrt3.
4. So the answer is \frac{3+\sqrt3}{4}.

Simplify \sqrt{20} first, then multiply the numerator and denominator by the denominator with its sign reversed.

1. Simplify the surd: \sqrt{20}=\sqrt{4\times5}=2\sqrt5, so the denominator is 2\sqrt5-3.
2. Multiply the numerator and denominator by 2\sqrt5+3. Denominator: (2\sqrt5)^2-3^2=20-9=11.
3. Numerator: (4+\sqrt5)(2\sqrt5+3)=8\sqrt5+12+2\times5+3\sqrt5=22+11\sqrt5.
4. So the expression is \frac{22+11\sqrt5}{11}. Divide both terms by 11: 2+\sqrt5.

Rationalise the given value before squaring it and subtracting it.

1. Rationalise u first: multiply the numerator and denominator by \sqrt{13}+1. The denominator is 13-1=12, so u=\frac{4(\sqrt{13}+1)}{12}=\frac{\sqrt{13}+1}{3}.
2. Square it: (\sqrt{13}+1)^2=13+2\sqrt{13}+1=14+2\sqrt{13}, so u^2=\frac{14+2\sqrt{13}}{9}.
3. To subtract, write u over 9 as well: u=\frac{3\sqrt{13}+3}{9}.
4. Subtract the numerators: (14+2\sqrt{13})-(3\sqrt{13}+3)=11-\sqrt{13}.
5. So u^2-u=\frac{11-\sqrt{13}}{9}.