Surds and Indices Concepts

1. Basic Surd Simplification

• Formula:

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

Extract perfect-square factors:

$$\sqrt{a^2b}=a\sqrt b$$

• Example: Simplify (\sqrt{72}). Solution:

$$\sqrt{72}=\sqrt{36\times2}$$ $$=6\sqrt2$$ $$\boxed{6\sqrt2}$$

√72
 ↓
√(36 × 2)
 ↓
6√2

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2. Addition & Subtraction of Surds

• Formula: Like surds can be combined:

$$a\sqrt{x}+b\sqrt{x}=(a+b)\sqrt{x}$$

Unlike surds cannot be directly added.

• Example: Simplify (3\sqrt5+2\sqrt5-\sqrt5). Solution:

$$(3+2-1)\sqrt5$$ $$=\boxed{4\sqrt5}$$

3√5 + 2√5 - √5
       ↓
(3 + 2 - 1)√5
       ↓
      4√5

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3. Multiplication & Division of Surds

• Formula:

$$\sqrt a\sqrt b=\sqrt{ab}$$ $$\frac{\sqrt a}{\sqrt b}=\sqrt{\frac ab}$$

• Example: Simplify (\sqrt6\times\sqrt{24}). Solution:

$$\sqrt{6\times24} =\sqrt{144} =\boxed{12}$$

√6 × √24
   ↓
√144
   ↓
  12

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4. Rationalizing a Single Surd Denominator

• Formula:

$$\frac{a}{\sqrt b} =\frac{a\sqrt b}{b}$$

• Example: Rationalize (\frac{5}{\sqrt3}). Solution: Multiply numerator and denominator by (\sqrt3):

$$\frac5{\sqrt3}\times\frac{\sqrt3}{\sqrt3}$$ $$=\frac{5\sqrt3}{3}$$ $$\boxed{\frac{5\sqrt3}{3}}$$

     5             5√3
   ────    →     ─────
    √3              3

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5. Rationalizing Binomial Surd Denominators

• Formula: Use the conjugate:

$$(a+\sqrt b)(a-\sqrt b)=a^2-b$$

Therefore:

$$\frac1{\sqrt a+\sqrt b} =\frac{\sqrt a-\sqrt b}{a-b}$$

• Example: Rationalize

$$\frac1{\sqrt5+\sqrt2}.$$

Solution: Multiply by the conjugate:

$$\frac1{\sqrt5+\sqrt2} \times \frac{\sqrt5-\sqrt2}{\sqrt5-\sqrt2}$$ $$=\frac{\sqrt5-\sqrt2}{5-2}$$ $$=\boxed{\frac{\sqrt5-\sqrt2}{3}}$$

  1          √5 - √2
──────  →   ─────────
√5 + √2         3

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6. Rationalizing Denominators with Two Surds and Coefficients

• Formula:

$$\frac1{a\sqrt b+c\sqrt d}$$

multiply numerator and denominator by the conjugate:

$$a\sqrt b-c\sqrt d$$

using:

$$(a\sqrt b)^2-(c\sqrt d)^2=a^2b-c^2d$$

• Example: Rationalize

$$\frac1{2\sqrt3+\sqrt2}.$$

Solution:

$$\frac1{2\sqrt3+\sqrt2} \times \frac{2\sqrt3-\sqrt2}{2\sqrt3-\sqrt2}$$

Denominator:

$$(2\sqrt3)^2-(\sqrt2)^2=12-2=10$$

Therefore:

$$\boxed{\frac{2\sqrt3-\sqrt2}{10}}$$

       1
───────────────
 2√3 + √2

multiply by:

 2√3 - √2
──────────
 2√3 - √2

        ↓

 2√3 - √2
───────────
    10

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7. Surd Comparison

• Formula: For positive quantities:

$$a>b\iff a^2>b^2$$

when comparing positive surds.

• Example: Which is larger, (2\sqrt3) or (3\sqrt2)? Solution: Square both:

$$(2\sqrt3)^2=12$$ $$(3\sqrt2)^2=18$$

Since:

$$18>12$$ $$\boxed{3\sqrt2>2\sqrt3}$$

2√3  → square → 12
3√2  → square → 18

18 > 12
Therefore 3√2 is larger.

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8. Comparing Surds by Squaring

• Formula: To compare expressions such as

$$\sqrt a+\sqrt b$$

square both sides when both sides are non-negative.

• Example: Which is larger: (\sqrt7+\sqrt2) or (\sqrt{10}+1)? Solution: Both are positive. Square them:

$$(\sqrt7+\sqrt2)^2=9+2\sqrt{14}$$

and:

$$(\sqrt{10}+1)^2=11+2\sqrt{10}$$

Since:

$$2\sqrt{14}>2\sqrt{10}$$

and the difference in the rational parts is only 2, check directly:

$$9+2\sqrt{14}\approx16.48$$ $$11+2\sqrt{10}\approx17.32$$

Therefore:

$$\boxed{\sqrt{10}+1>\sqrt7+\sqrt2}$$

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9. Rationalizing Complex Nested Denominators

• Formula: Rationalize in stages, using conjugates:

$$(a+b)(a-b)=a^2-b^2$$

• Example: Simplify

$$\frac1{2+\sqrt3}.$$

Solution: Multiply by the conjugate (2-\sqrt3):

$$\frac1{2+\sqrt3}\times\frac{2-\sqrt3}{2-\sqrt3}$$ $$=\frac{2-\sqrt3}{4-3}$$ $$=\boxed{2-\sqrt3}$$

       1
────────────
  2 + √3

   conjugate
      ↓
  2 - √3

       ↓

  2 - √3
──────────
     1

= 2 - √3

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10. Laws of Indices

• Formula:

$$a^m\times a^n=a^{m+n}$$ $$\frac{a^m}{a^n}=a^{m-n}$$ $$(a^m)^n=a^{mn}$$ $$(ab)^n=a^nb^n$$

• Example: Simplify

$$\frac{x^7\times x^3}{x^4}.$$

Solution:

$$=x^{7+3-4}$$ $$=\boxed{x^6}$$

x⁷ × x³
───────
 x⁴

   ↓

x^(7+3-4)

   ↓

  x⁶

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11. Zero & Negative Indices

• Formula:

$$a^0=1\quad(a\ne0)$$ $$a^{-n}=\frac1{a^n}$$

• Example: Simplify

$$2^{-3}+5^0.$$

Solution:

$$2^{-3}=\frac18$$

and:

$$5^0=1$$

Therefore:

$$\frac18+1 =\boxed{\frac98}$$

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12. Fractional Indices

• Formula:

$$a^{1/n}=\sqrt[n]{a}$$ $$a^{m/n}=\sqrt[n]{a^m}$$

• Example: Simplify (27^{2/3}). Solution:

$$27^{2/3}=(\sqrt[3]{27})^2$$ $$=3^2$$ $$\boxed9$$

27^(2/3)
   ↓
(∛27)²
   ↓
  3²
   ↓
   9

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13. Mixed Indices and Surds

• Formula:

$$a^{1/2}=\sqrt a,\qquad a^{m/n}=\sqrt[n]{a^m}$$

• Example: Simplify

$$16^{3/4}.$$

Solution:

$$16^{3/4} =(\sqrt[4]{16})^3$$ $$=2^3$$ $$\boxed8$$

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14. Scientific Notation

• Formula:

$$N=a\times10^n,\qquad1\leq a<10$$

Moving the decimal left gives a positive exponent; moving it right gives a negative exponent.

• Example: Express (0.000072) in scientific notation. Solution: Move the decimal 5 places to the right:

$$0.000072=7.2\times10^{-5}$$ $$\boxed{7.2\times10^{-5}}$$

0.000072
   ↓ move decimal 5 places →
7.2 × 10⁻⁵

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15. Operations in Scientific Notation

• Formula:

$$(a\times10^m)(b\times10^n) =ab\times10^{m+n}$$ $$\frac{a\times10^m}{b\times10^n} =\frac ab\times10^{m-n}$$

• Example: Calculate

$$(3\times10^5)(2\times10^3).$$

Solution:

$$=6\times10^{5+3}$$ $$=\boxed{6\times10^8}$$

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16. Simplification of Mixed Algebraic Indices

• Formula:

$$(a^mb^n)^p=a^{mp}b^{np}$$

Apply the power to every factor inside the bracket.

• Example: Simplify

$$(2x^{-2}y^3)^3\div(4x^4y^{-1}).$$

Solution: First expand:

$$(2x^{-2}y^3)^3 =8x^{-6}y^9$$

Therefore:

$$\frac{8x^{-6}y^9}{4x^4y^{-1}}$$ $$=2x^{-10}y^{10}$$

Using (x^{-10}=1/x^{10}):

$$\boxed{\frac{2y^{10}}{x^{10}}}$$

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Advanced Variants

17. Infinite Nested Radicals

• Formula: For

$$x=\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}},$$

set:

$$x=\sqrt{a+x}$$

so:

$$x^2-x-a=0$$

and take the positive root:

$$x=\frac{1+\sqrt{1+4a}}2$$

• Example: Evaluate

$$\sqrt{12+\sqrt{12+\sqrt{12+\cdots}}}.$$

Solution: Let:

$$x=\sqrt{12+x}$$

Then:

$$x^2=12+x$$ $$x^2-x-12=0$$ $$(x-4)(x+3)=0$$

Since (x>0):

$$\boxed{x=4}$$

x = √(12 + √(12 + √(12 + ...)))

Since the inside repeats:

x = √(12 + x)

x² = 12 + x
x² - x - 12 = 0

(x - 4)(x + 3) = 0

x = 4

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18. Infinite Nested Radical with a Different Constant

• Formula:

$$x=\sqrt{a+bx}$$

gives:

$$x^2-bx-a=0$$

Take the positive root.

• Example: Evaluate

$$\sqrt{6+2\sqrt{6+2\sqrt{6+\cdots}}}.$$

Solution: Let:

$$x=\sqrt{6+2x}$$

Then:

$$x^2=6+2x$$ $$x^2-2x-6=0$$ $$x=1+\sqrt7$$

(the other root is negative).

$$\boxed{1+\sqrt7}$$