Summary of indices

Introduction to Indices

Indices, also known as powers or exponents, deal with multiplying a number by itself for a specified number of times.
The base number is the number being multiplied, and the index indicates how many times the base number is multiplied.
For an example, 5^3 – the base is 5 and the index is 3. This means 5 is multiplied by itself 3 times resulting in 125.

Understanding Indices

Zero as an index: Any number (except zero) to the power of zero is always 1. For example, 2^0 = 1.
Negative indices: A negative index indicates a reciprocal. For example, 2^-1 equals 1/2.
Fractional indices: The numerator of the fraction indicates the power, while the denominator represents the root. For example, 2^(1/2) equals to square root of 2.

Indices Laws

Multiplication Law: When multipying numbers with the same base, the exponents are added. Example: a^b * a^c = a^(b+c).
Division Law: When dividing numbers with the same base, the exponents are subtracted. Example: a^b / a^c = a^(b-c).
Power Law: When raising a power to another power, the exponents are multiplied. Example: (a^b)^c = a^(b*c).
Zero Powers Law: Any number (except zero) to the power of zero equals 1. Example: a^0 = 1, if a ≠ 0.
Negative Power Law: A negative power means to take the reciprocal. Example: a^-b = 1/a^b.
Fractional Power Law: A fractional exponent is the same as a root. Example: a^(1/b) = ^b√a.

Solving Problems with Indices

When solving equations involving indices, often the first step is to simplify the equation using the laws of indices.
Following the simplification, standard methods for solving linear and quadratic equations can be employed.

Conclusion

Indices offer a concise way of expressing repeated multiplication of a fixed number.
Understanding indices and the associated laws of multiplication, division, powers, zero and negatives is a necessary foundation for further algebra problems.
Practice with a variety of problems involving indices will enhance algebraic problem-solving skills.