Surds and Indices

Review basic laws of surds and indices

1 Surds

• A surd is an irrational number expressed in terms of roots (square, cube, etc.) that cannot be simplified.

• Examples: $\sqrt{2},\sqrt{3},\sqrt[3]{6}$.

• $\sqrt{9}$ is not a surd because it can be further simplified to $3$, which is a rational number.

• Types:

  • Pure: have a single irrational number, e.g., $\sqrt{2},\sqrt{3},\sqrt[3]{6}$.

  • Mixed: have rational and irrational numbers, e.g., $3\sqrt{2}$.

1.1 Laws of surds

1. $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$

2. $\sqrt{a}\times \sqrt{a}=a$

3. ${\displaystyle \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}}$

4. $\sqrt{a}\pm \sqrt{b}\ne \sqrt{a\pm b}$

5. ${\displaystyle \sqrt[n]{a}=a^{\frac{1}{n}}}$

Example M.4.1

Simplify $\sqrt{28}$

$$\sqrt{28}=$$

$$\sqrt{4\times 7}=$$

$$\sqrt{4}\times \sqrt{7}=$$

$$2\sqrt{7}$$

Example M.4.2

Simplify ${\displaystyle \sqrt{\frac{52}{9}}}$

$$\sqrt{\frac{52}{9}}=$$

$$\frac{\sqrt{52}}{\sqrt{9}}=$$

$$\frac{\sqrt{4\times 13}}{3}=$$

$$\frac{\sqrt{4}\times \sqrt{13}}{3}=$$

$$\frac{2}{3}\sqrt{13}$$

Exercise M.4.1

For each of the following roots, indicate whether it is surd or not:

$\sqrt{25}$ surdnot a surd

$\sqrt{35}$ surdnot a surd

$\sqrt{67}$ surdnot a surd

$\sqrt{144}$ surdnot a surd

0 of 4 correct

2 Indices

• An index (exponent) is the power to which a number (base) is raised.

• It indicates how many times the base is multiplied by itself.

• For example $4^2$, $4$ is the base and $2$ is the index (it is read as “four squared” or “four raised to the power two”).

2.1 Laws of indices

1. $a^m\times a^n=a^{m+n}$

2. ${\displaystyle \frac{a^m}{a^n}=a^{m-n}}$

3. ${\displaystyle a^{-m}=\frac{1}{a^m}}$

4. $(a^m{)}^n=a^{mn}$

5. ${\left(a^m{b}^n\right)}^k=a^{mk}\times b^{nk}$

6. $a^1=a$

7. $a^0=1$

Example M.4.3

Simplify ${\displaystyle \frac{x^6}{x^2}}$

$$\frac{x^6}{x^2}=x^{6-2}=x^4$$

Exercise M.4.2

Simplify $y^3(y^{-2}-y^{-3})$

The answer is

3 References

• Basic Algebra. Help Engineers Learn Mathematics (HELM) workbooks. Loughborough University. Retrieved August 14, 2024, from https://www.lboro.ac.uk/media/media/schoolanddepartments/mlsc/downloads/Basic%20Algebra.pdf

• Surds. Cuemath. Retrieved August 14, 2024, from https://www.cuemath.com/numbers/surds/

• Index Laws Revision. Maths Learning Centre. University of Adelaide. Retrieved August 14, 2024, from https://www.adelaide.edu.au/mathslearning/ua/media/24/index-laws-revision.pdf

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