Surds, roots and irrational numbersIB MYP Maths Standard: Revision notes

The square root $\sqrt{a}$ is the positive number that gives $a$ when multiplied by itself: $\sqrt{49}=7$ because $7\times7=49$. The cube root $\sqrt[3]{a}$ is the number that gives $a$ when multiplied by itself three times: $\sqrt[3]{64}=4$ because $4\times4\times4=64$. Learn the square numbers $1,4,9,16,25,36,49,64,81,100,121,144$ and the cubes $1,8,27,64,125$. Square-rooting is the inverse of squaring, so $(\sqrt{a})^2=a$.

A rational number can be written as a fraction of two integers, such as $\frac34$ or $0.25$ or $5$. An irrational number cannot. Its decimal goes on forever without repeating. A surd is a root that is irrational, such as $\sqrt2$, $\sqrt{20}$ or $\sqrt[3]{5}$. $\sqrt{25}=5$ is not a surd because it is a whole number, but $\sqrt{26}$ is a surd. $\pi$ is irrational too, but it is not a surd. A surd is an exact value; $\sqrt2\approx1.414$ is only an approximation.

Use $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$. Look for the largest square factor of the number under the root, take its square root and leave the rest. Example: $\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt2=5\sqrt2$. Example: $\sqrt{18}=\sqrt{9\times2}=3\sqrt2$ and $\sqrt{72}=\sqrt{36\times2}=6\sqrt2$. If you pick a smaller square factor, you can simplify again: $\sqrt{72}=\sqrt{4\times18}=2\sqrt{18}=2\times3\sqrt2=6\sqrt2$.

You can only add or subtract like surds, which have the same number under the root. Treat $\sqrt2$ like a letter in algebra: $3\sqrt2+2\sqrt2=5\sqrt2$. Simplify each surd first to see whether they are alike: $\sqrt{18}+\sqrt8=3\sqrt2+2\sqrt2=5\sqrt2$ and $\sqrt{18}-\sqrt8=\sqrt2$. Unlike surds cannot be combined: $\sqrt2+\sqrt3$ stays as it is.

Multiply the numbers outside the roots together and the numbers inside the roots together: $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$ and $(m\sqrt a)(n\sqrt b)=mn\sqrt{ab}$. Examples: $\sqrt3\times\sqrt{12}=\sqrt{36}=6$, $(3\sqrt2)(2\sqrt2)=6\times2=12$, and $(\sqrt5)^2=5$. To expand brackets, multiply every term: $(3+\sqrt5)(3-\sqrt5)=9-3\sqrt5+3\sqrt5-5=4$. Notice that the surd terms cancelled.

Leave answers as surds when the question says exact. Only convert to a decimal at the end, using a calculator, and give a suitable number of significant figures. Rounding early introduces errors: with $\sqrt2\approx1.4$, a total of $20\sqrt2$ is $28$, but the true value is $28.28\ldots$. When a length must be long enough, such as fencing or a shelf, round up.