Recall that to simplify an expression means to rewrite it by combining terms or indices. To simplify expressions involving indices, we use the properties of indices. The properties of indices may be combined to simplify expressions.

Doing the group work 1 will help you to understand the properties of indices to simplify expressions involving indices.

Consider the product p³ × p⁴. Both terms have the same base, p, but they are raised to different exponents. Expand each expression and then rewrite the resulting expression as follows:

The result is that p³ × p⁴ = p³⁺⁴ = p⁷

The Product Property of Indices

For any real number a and natural numbers m and n, the product property of indices states that

a^m × aⁿ = a^m+n

The quotient Properties of indices allows us to simplify an expression that divides two numbers with the same base but different indices. We can simplify an expression

The Quotient Property of Indices

For any real number a and natural numbers m and n such that m > n, the quotient property of indices states that

Suppose an indicial expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of indices. Consider the expression (a³)². The expression inside the parentheses is multiplied three times because it has an index of 3. Then, the result is multiplied four times because the given expression has an index of 4.

The Power Property of Indices

For any real number a and natural numbers m and n, the power property of indices states that

(a^m)ⁿ = a^m×n

The Power of a Product Property of Indices

For any real numbers a and b and a natural number n, the power of a product property of indices states that

(ab)ⁿ = aⁿbⁿ

The Zero Index Property of Indices

For any nonzero real numbers a, the zero-index property of indices states that

a⁰ = 1.