4.6 Surds, Indices & Simplification

Laws of Indices

These eight rules generate every indices question at this level.

$$a^{m} \times a^{n} = a^{m+n} \qquad \frac{a^{m}}{a^{n}} = a^{m-n}$$ $$(a^{m})^{n} = a^{mn} \qquad (ab)^{m} = a^{m}b^{m}$$ $$\left(\frac{a}{b}\right)^{m} = \frac{a^{m}}{b^{m}} \qquad a^{0} = 1 \;\; (a \neq 0)$$ $$a^{-m} = \frac{1}{a^{m}} \qquad a^{m/n} = \sqrt[n]{a^{m}}$$

The rule $a^0 = 1$ holds for every non-zero base, and the expression $0^0$ is undefined — a distinction that occasionally appears as a stand-alone item.

Solving exponential equations

The standard technique is to force a common base and then equate exponents.

Example. Solve $2^{x+3} = 32$.

Since $32 = 2^5$, we need $x + 3 = 5$, so $x = 2$.

Example. Solve $9^{x} = 27^{x-1}$.

Write both as powers of 3: $3^{2x} = 3^{3(x-1)}$, so $2x = 3x - 3$ and $x = 3$.

Example. If $a^{x} = b^{y} = c^{z}$ and $b^2 = ac$, show the relation among the exponents. Taking logarithms gives $x\log a = y\log b = z\log c = k$. From $b^2 = ac$ we get $2\log b = \log a + \log c$, so $\dfrac{2k}{y} = \dfrac{k}{x} + \dfrac{k}{z}$, giving

$$\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$$

which says $y$ is the harmonic mean of $x$ and $z$.

Comparing surds and powers

To compare quantities such as $\sqrt{2}$, $\sqrt[3]{3}$ and $\sqrt[4]{5}$, raise all to a common power — here the LCM of 2, 3 and 4, which is 12:

$$(\sqrt{2})^{12} = 2^{6} = 64, \quad (\sqrt[3]{3})^{12} = 3^{4} = 81, \quad (\sqrt[4]{5})^{12} = 5^{3} = 125$$

So the ascending order is $\sqrt{2} < \sqrt[3]{3} < \sqrt[4]{5}$.

Surds

A surd is an irrational root that cannot be simplified to a rational number, such as $\sqrt{2}$ or $\sqrt[3]{5}$. Note that $\sqrt{9} = 3$ is not a surd because it resolves to a rational value.

Basic operations

$$\sqrt{a}\times\sqrt{b} = \sqrt{ab}, \qquad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$$

Surds add only when they are like surds: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$, but $\sqrt{2} + \sqrt{3}$ cannot be combined.

Always simplify first: $\sqrt{50} = \sqrt{25\times2} = 5\sqrt{2}$, so $\sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$.

Rationalising the denominator

Multiply numerator and denominator by the conjugate of the denominator. The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$, and their product is the rational number $a^2 - b$.

Example.

$$\frac{1}{3 + \sqrt{5}} = \frac{1}{3+\sqrt5}\times\frac{3-\sqrt5}{3-\sqrt5} = \frac{3-\sqrt5}{9-5} = \frac{3-\sqrt5}{4}$$

Example.

$$\frac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3} = \frac{(\sqrt7+\sqrt3)^2}{7-3} = \frac{7+3+2\sqrt{21}}{4} = \frac{10+2\sqrt{21}}{4} = \frac{5+\sqrt{21}}{2}$$

Values worth memorising

$$\sqrt2 \approx 1.414, \quad \sqrt3 \approx 1.732, \quad \sqrt5 \approx 2.236, \quad \sqrt6 \approx 2.449, \quad \sqrt7 \approx 2.646$$

These convert a surd expression into a numerical estimate in seconds, which is often all a multiple-choice item requires.

Simplification and BODMAS

The order of operations is:

1. Brackets, innermost first: $(\;)$ then $[\;]$ then $\{\;\}$
2. Orders — powers and roots
3. Division and Multiplication, left to right
4. Addition and Subtraction, left to right

The two steps most often mishandled are that division and multiplication have equal priority, evaluated left to right, and likewise addition and subtraction. Treating multiplication as automatically preceding division produces wrong answers.

Example. Evaluate $48 \div 6 \times 2 + 5$.

Working left to right through the division and multiplication: $48 \div 6 = 8$, then $8 \times 2 = 16$, then $16 + 5 = 21$. Answering 9, by doing $6 \times 2$ first, is the standard error.

Approximation Technique

With 100 Paper-I questions sharing a three-hour window with 100 engineering questions, exact computation is often wasteful. Two habits pay off:

Round sensibly and track the direction of error. To estimate $\dfrac{4986 \times 39}{198}$, round to $\dfrac{5000 \times 40}{200} = 1000$. Since the numerator was rounded up and the denominator up as well, the true value is close to but slightly below 1000 — enough to identify the option.

Use fraction equivalents. To find 37.5% of 216, recognise 37.5% as $3/8$, giving $216 \times 3 / 8 = 81$ directly.

Check the last digit. When options differ in their final digit, computing only the units digit identifies the answer. For $47 \times 63$, the units digit is that of $7 \times 3 = 21$, namely 1.

Because there is no negative marking, an approximation that narrows four options to one is worth exactly as much as an exact computation, and costs a fraction of the time.