Introduction to indices (exponents)

Introduction to Indices (Exponents)

• Indices are also known as exponents or powers in mathematics. They represent the number of times a number or expression is multiplied by itself.
• For example, in the expression ‘2^3’, the base is ‘2’ and the exponent or index is ‘3’. The expression is calculated as ‘2 * 2 * 2’, which equals ‘8’.
• The base number can also be a letter representing a variable, such as ‘x’. For example, ‘x^2’ means ‘x * x’.

Working with Indices

• Rule 1: When multiplying terms with the same base, keep the base the same and add the exponents together. For example, ‘x^2 * x^3’ becomes ‘x^(2+3)’ or ‘x^5’.
• Rule 2: When dividing terms with the same base, keep the base the same and subtract the exponents. For example, ‘x^5 / x^2’ becomes ‘x^(5-2)’ or ‘x^3’.
• Rule 3: The exponent of 1 means the base number remains the same. For example, ‘x^1’ is ‘x’.
• Rule 4: The exponent of 0 equals 1, regardless of what the base number is. For example, ‘x^0’ is 1.

Special Indices

• Negative indices: A term with a negative index is equivalent to 1 divided by the term with the positive index. For example, ‘x^-2’ equals ‘1/x^2’.
• Fractional indices: A term with a fractional index, such as ‘x^(1/2)’, is equivalent to the square root of x.

Examples

• Simplifying ‘x^3 * x^4’ by adding the exponents gives ‘x^7’.
• Simplifying ‘x^5 / x^2’ by subtracting the exponents becomes ‘x^3’.
• Simplifying ‘x^-2’ by changing the negative exponent to a positive exponent becomes ‘1/x^2’.
• Simplifying ‘x^(1/2)’ by changing the fractional exponent to a root becomes the square root of x or ‘√x’.

Conclusion

• Understanding how to work with indices (exponents) is a fundamental skill used frequently in algebra.
• These rules apply whether the base is a number or a variable. They are crucial for simplifying expressions as well as solving equations.