CBSE Class 10 Mathematics Surds and Indices Notes Set 18

Radicals and Surds

A radical is an expression of the form \( \sqrt[n]{x} \), where n is the order and x is the radicand; it is the same as \( x^{1/n} \).
If a is a positive rational number that is not the nth power of any rational number, then \( \sqrt[n]{a} \) is a surd. Every surd is irrational, but not every irrational number is a surd (for example \( \pi \) is not).

Types of Surds

1. Pure surd: only a surd, like \( \sqrt7 \) or \( \sqrt[3]{6} \).
2. Mixed surd: a rational number combined with a surd, like \( 2+\sqrt3 \).
3. Similar (like) surds: different rational multiples of the same surd, like \( 2\sqrt2,\ 5\sqrt2,\ 7\sqrt2 \).
4. Binomial surd: the sum or difference of two surds, like \( \sqrt6+4\sqrt5 \).
5. Equality: \( a+b\sqrt{x}=c+d\sqrt{x} \) is true only when a = c and b = d.

Comparing Surds

Surds of the same order are compared through their radicands. For different orders, first convert all of them to the LCM of the orders and then compare the radicands.
Example: to compare \( \sqrt[4]{6} \) and \( \sqrt[3]{5} \), write them as \( \sqrt[12]{6^3}=\sqrt[12]{216} \) and \( \sqrt[12]{5^4}=\sqrt[12]{625} \); hence the second is larger.

Laws of Radicals and Indices

\( a^m\times a^n=a^{m+n} \),  \( a^m\div a^n=a^{m-n} \),  \( (a^m)^n=a^{mn} \),  \( a^0=1 \),  \( a^{-n}=\dfrac1{a^n} \)
\( \sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab} \),  \( \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac ab} \),  \( \sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a} \),  \( a^{m/n}=\sqrt[n]{a^m} \)
\( (ab)^n=a^nb^n \),  \( \left(\dfrac ab\right)^{n}=\dfrac{a^n}{b^n} \).
Equal bases: if \( a^x=a^y \) (a > 0, a ≠ 1) then x = y.

Squares of Surd Expressions

\( (\sqrt a+\sqrt b)^2=a+b+2\sqrt{ab} \) and \( (\sqrt a-\sqrt b)^2=a+b-2\sqrt{ab} \).
Square root of a binomial surd: to find \( \sqrt{p+2\sqrt q} \), look for two numbers a and b with a + b = p and ab = q; then the result is \( \sqrt a+\sqrt b \) (for the minus sign use the positive difference).

Rationalisation

Multiplying numerator and denominator by a suitable factor so that the denominator becomes rational is called rationalisation.
\( \dfrac1{\sqrt a+\sqrt b}=\dfrac{\sqrt a-\sqrt b}{a-b} \),  \( \dfrac1{\sqrt a-\sqrt b}=\dfrac{\sqrt a+\sqrt b}{a-b} \).
For cube roots use \( a^3+b^3=(a+b)(a^2-ab+b^2) \) and \( a^3-b^3=(a-b)(a^2+ab+b^2) \). For example the rationalising factor of \( \sqrt[3]{a}-\sqrt[3]{b} \) is \( \sqrt[3]{a^2}+\sqrt[3]{ab}+\sqrt[3]{b^2} \).
Nested root: if x = n(n + 1), then \( \sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}=n+1 \) and \( \sqrt{x-\sqrt{x-\sqrt{x-\cdots}}}=n \).

Useful Index Results

1. If \( a+b+c=0 \) then \( a^3+b^3+c^3=3abc \).
2. If \( x+\dfrac1x=a \) then \( x^3+\dfrac1{x^3}=a^3-3a \).
3. If \( a^x=b^y=c^z=k \) then \( a=k^{1/x},\ b=k^{1/y},\ c=k^{1/z} \); if also abc = 1 then \( \dfrac1x+\dfrac1y+\dfrac1z=0 \).
4. Write every term with a common base before equating exponents.

Worked Examples

Illustration 1. Simplify \( \dfrac{5+\sqrt3}{2-\sqrt3} \).
Answer:
Multiply by \( \dfrac{2+\sqrt3}{2+\sqrt3} \): numerator \( 10+5\sqrt3+2\sqrt3+3=13+7\sqrt3 \), denominator 4 − 3 = 1.
Answer: \( 13+7\sqrt3 \).

Illustration 2. Find \( \sqrt{7+4\sqrt3} \).
Answer:
\( 7+4\sqrt3=7+2\sqrt{12} \), and 4 + 3 = 7, 4 × 3 = 12.
So \( \sqrt{7+4\sqrt3}=2+\sqrt3 \).

Illustration 3. If \( 2^x=4^y=8^z \) and xyz = 288, find x, y, z.
Answer:
Then x = 2y = 3z. Put x = 6k, y = 3k, z = 2k.
36k³ = 288 gives k = 2, so x = 12, y = 6, z = 4.

Illustration 4. Solve \( 4^x-4^{x-1}=24 \).
Answer:
\( 4^{x-1}(4-1)=24 \), so \( 4^{x-1}=8 \).
\( 2^{2x-2}=2^3 \) gives x = 5/2.