Surds and Indices

Review basic laws of surds and indices
Published

August 14, 2024


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. \(\displaystyle \sqrt[n]{a} = a^{\frac{1}{n}}\)

Simplify \(\sqrt{28}\)

\[ \sqrt{28}= \]

\[\sqrt{4\times 7} = \]

\[ \sqrt{4} \times \sqrt{7} = \]

\[ 2\sqrt{7} \]

Simplify \(\displaystyle \sqrt{\frac{52}{9}}\)

\[ \sqrt{\frac{52}{9}} = \]

\[ \frac{\sqrt{52}}{\sqrt{9}} = \]

\[ \frac{\sqrt{4 \times 13}}{3} = \]

\[ \frac{\sqrt{4} \times \sqrt{13}}{3} = \]

\[ \frac{2}{3}\sqrt{13} \]

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

\(\sqrt{25}\)

\(\sqrt{35}\)

\(\sqrt{67}\)

\(\sqrt{144}\)

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

  6. \(a^1=a\)

  7. \(a^0=1\)

Simplify \(\displaystyle \frac{x^6}{x^2}\)

\[ \frac{x^6}{x^2} = x^{6-2} = x^4 \]

Simplify \(y^3(y^{-2}-y^{-3})\)

The answer is

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