Understanding Index Laws

• Index Laws

Multiplication

• When multiplying terms with the same base, add the exponents: (a^m \times a^n = a^{m+n}).
• Example: (2^3 \times 2^2 = 2^{3+2} = 2^5 = 32).

Division

• When dividing terms with the same base, subtract the exponents: (a^m \div a^n = a^{m-n}).
• Example: (3^5 \div 3^2 = 3^{5-2} = 3^3 = 27).

Power of a Power

• When raising a power to another power, multiply the exponents: ((a^m)^n = a^{m \times n}).
• Example: ((5^2)^3 = 5^{2 \times 3} = 5^6 = 15625).

Zero Exponent

• Any non-zero number raised to the power of zero is 1: (a^0 = 1) (where (a \neq 0)).
• Example: (7^0 = 1).

Negative Exponent

• A negative exponent indicates a reciprocal: (a^{-n} = \frac{1}{a^n}).
• Example: (4^{-2} = \frac{1}{4^2} = \frac{1}{16}).

Fractional Exponent

• A fractional exponent ( \frac{1}{n} ) indicates the ( n )th root: ( a^{\frac{1}{n}} = \sqrt[n]{a} ).

• Example: ( 8^{\frac{1}{3}} = \sqrt[3]{8} = 2 ).

• A fractional exponent ( \frac{m}{n} ) indicates both a power and a root: ( a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m ).

• Example: ( 9^{\frac{3}{2}} = \sqrt{9^3} = (\sqrt{9})^3 = 3^3 = 27 ).

• Standard Form

Definition

• Standard form (also known as scientific notation) expresses numbers as (A \times 10^n), where (1 \leq |A| < 10) and (n) is an integer.

Expressing Large Numbers

• Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point. The number of places moved is the value of (n).
• Example: (54300 = 5.43 \times 10^4).

Expressing Small Numbers

• Move the decimal point to the right until there is one non-zero digit to the left of the decimal point. The number of places moved is the value of (n), but it will be negative.
• Example: (0.0023 = 2.3 \times 10^{-3}).

Calculations

• When multiplying numbers in standard form, multiply the (A) values and add the exponents.

• Example: ((2 \times 10^3) \times (3 \times 10^4) = (2 \times 3) \times 10^{3+4} = 6 \times 10^7).

• When dividing numbers in standard form, divide the (A) values and subtract the exponents.

• Example: ((8 \times 10^5) \div (2 \times 10^2) = (8 \div 2) \times 10^{5-2} = 4 \times 10^3).

• Rational and Irrational Numbers

Rational Numbers

• Definition: Numbers that can be expressed in the form ( \frac{p}{q} ), where ( p ) and ( q ) are integers and ( q \neq 0 ).
• Examples: ( \frac{1}{2} ), ( -3 ), ( 0.75 ) (which can be written as ( \frac{3}{4} )), ( \sqrt{4} = 2 ).
• Decimal representation: Rational numbers have decimal expansions that either terminate or repeat.
• Terminating decimals: ( 0.25 )
• Repeating decimals: ( 0.333... = 0.\overline{3} )

Irrational Numbers

• Definition: Numbers that cannot be expressed in the form ( \frac{p}{q} ), where ( p ) and ( q ) are integers.
• Examples: ( \sqrt{2} ), ( \pi ), ( e ).
• Decimal representation: Irrational numbers have non-terminating, non-repeating decimal expansions.

Key Differences

• Rational numbers can be written as a simple fraction; irrational numbers cannot.

• Rational numbers have decimal expansions that either terminate or repeat; irrational numbers do not.

• Simplifying Surds and Solving Exact Roots Using Prime Factorisation

Simplifying Surds

• Surd Definition: A surd is an irrational number expressed as a root.
• Basic Simplification: Look for perfect square factors (or perfect cube, etc., depending on the root) within the surd.
• Example: ( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} ).
• General Form: ( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} )

Simplifying Surds with Larger Numbers

• Prime Factorisation: Break down the number under the square root into its prime factors.
• Example: Simplify ( \sqrt{180} ).
• Prime factorisation of 180: ( 180 = 2^2 \times 3^2 \times 5 ).
• ( \sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{5} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5} ).

Solving Exact Roots Using Prime Factorisation

• Prime Factorisation Method: Use prime factorisation to find exact roots, particularly for cube roots, fourth roots, etc.
• Example: Find ( \sqrt[3]{216} ).
• Prime factorisation of 216: ( 216 = 2^3 \times 3^3 ).
• ( \sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3} = \sqrt[3]{2^3} \times \sqrt[3]{3^3} = 2 \times 3 = 6 ).

Example with Higher Root

• Find ( \sqrt[4]{625} ).

• Prime factorisation of 625: ( 625 = 5^4 ).

• ( \sqrt[4]{625} = \sqrt[4]{5^4} = 5 ).

• Changing Recurring Decimals to Fractions

Recurring Decimals

• Definition: A recurring decimal (or repeating decimal) is a decimal number that has a repeating sequence of digits after the decimal point.

Method

• Let ( x ) be the recurring decimal.
• Multiply ( x ) by a power of 10 (e.g., 10, 100, 1000) such that the repeating part lines up.
• ( 10x, 100x, 1000x )
• Subtract ( x ) (or a multiple of ( x )) from this new equation to eliminate the repeating part.
• Solve for ( x ) to express as a fraction.

Simple Recurring Decimal Example

• Convert ( 0.\overline{3} ) to a fraction.
• Let ( x = 0.333... )
• Multiply by 10: ( 10x = 3.333... )
• Subtract ( x ) from ( 10x ): ( 10x - x = 3.333... - 0.333... )
• ( 9x = 3 )
• ( x = \frac{3}{9} = \frac{1}{3} )

More Complex Recurring Decimal Example

• Convert ( 0.\overline{27} ) to a fraction.
• Let ( x = 0.272727... )
• Multiply by 100: ( 100x = 27.272727... )
• Subtract ( x ) from ( 100x ): ( 100x - x = 27.272727... - 0.272727... )
• ( 99x = 27 )
• ( x = \frac{27}{99} = \frac{3}{11} )

Example with Non-Repeating Part

• Convert ( 0.1\overline{6} ) to a fraction.
• Let ( x = 0.1666... )
• Multiply by 10: ( 10x = 1.666... )
• Multiply by 100: ( 100x = 16.666... )
• Subtract ( 10x ) from ( 100x ): ( 100x - 10x = 16.666... - 1.666... )
• ( 90x = 15 )
• ( x = \frac{15}{90} = \frac{1}{6} )