Solution of Index Triangles

Solution of Index Triangles

Definition and Calculation

• Indices, also known as powers or exponents, indicate the number of times a base number is multiplied by itself.
• The base is the number that is multiplied, while the exponent indicates the number of times the base is used as a factor.
• For example, in 4^3, 4 is the base and 3 is the exponent, meaning 4 is multiplied by itself three times.

Key Rules for Indices

• Multiplication: When multiplying numbers with the same base, add the exponents: a^n * a^m = a^(n+m).
• Division: When dividing numbers with the same base, subtract the exponents: a^n / a^m = a^(n-m).
• Power: When raising an exponent to another power, multiply the exponents: (a^n)^m = a^(n*m).

Zero and Negative Indices

• For any nonzero number a, a^0 equals 1.
• Negative exponent means the reciprocal of the base raised to the positive power: a^(-n) equals 1/a^n.

Fractional Indices

• Fractional indices correspond to roots: a^(1/n) equals the nth root of a. For example, 8^(1/3) is the cube root of 8, which is 2.
• Mixed indices can be written as a mixture of a whole number and a fraction: 8^(5/3) is the cube root of 8 to the power 5.

Indices in Algebra

• Indices can also be combined with algebraic expressions.
• The same rules apply with x^n * x^m = x^(n+m), (x^n)^m = x^(n*m), and x^n / x^m = x^(n-m).

Remember that understanding the rules of indices helps simplify calculations, solve equations, and better understand exponential and logarithmic functions.