Mathematics

Algebra and Functions (Pure Mathematics)

# Indices

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Indices

Basic Rules of Indices

- Any number raised to the power of 1 is the number itself, e.g:
**a^1 = a** - Any number raised to the power of 0 is 1, e.g:
**a^0 = 1** - If two terms with the same base are multiplied together, the powers are added, e.g:
**a^m * a^n = a^(m+n)** - If a term with a base of "a" raised to a power "m" is itself raised to a power "n", multiply the powers, e.g:
**(a^m)^n = a^(mn)** - If two terms with the same base are divided, subtract the exponent of the denominator from the exponent of the numerator, e.g:
**a^m / a^n = a^(m-n)** - If a term in the denominator has a negative exponent, it can be moved to the numerator and made positive, e.g:
**1 / a^-n = a^n** - The
**n-th root**of a number "a" can be denoted by a ^ (1/n)

Laws of Indices Involving Fractions

**a^-n = 1 / a^n**, which means a reciprocal of a number can be written as a negative exponent.- The
**n-th root**of a number can be expressed as a power with a fractional exponent, e.g:**n√a = a^(1/n)** - If the power of a term is a fraction, the denominator of the fraction is the root, and the numerator is the power. For example,
**a^(m/n) = ( n√a ) ^m**

Simplification Using Indices

- In order to simplify expressions with indices, use the laws of indices to combine terms.
- Simplify the expression step by step, until no further simplification is possible.
- When dealing with algebraic expressions, it's important to remember that these rules apply only to terms with the same base.

Indices and Surds

- A Surd is an expression that includes a root (√). Surds are dealt with using the laws of indices.
- The method of 'rationalising the denominator' is used to remove surds from the denominator of a fraction.
- To simplify a surd, look for the largest square number which divides into the number under the surd, and use the rule
**√ab = √a * √b**.

Exponential Equations

- Exponential equations are those where the variable is in the exponent.
- They can be solved using the principle that if
**a^m = a^n**, then**m = n**. - In cases where this cannot be applied directly, logarithms may be used to solve the equation.