What is the Quotient law of exponents?

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The Quotient law of exponents states that,

For any non-zero integer a, 𝑎^𝑚 ÷ 𝑎^𝑛 = 𝑎^𝑚−𝑛, where m and n are whole numbers and m > n.

For example,

3⁵ ÷ 3³ = 3⁵⁻³ = 3².

This law applies to only the division of exponential forms with the same base.

Variation of quotient law when m < n

For any non-zero integer a, 𝑎^𝑚 ÷ 𝑎^𝑛 = 1/𝑎𝑛−𝑚, where m and n are whole numbers and m < n.

For example,

3³ ÷ 3⁵ = 1/35−3 = 1/3²

Problems based on quotient law

Express 𝟔^𝟗 ÷ 𝟔^𝟑in exponential form. Solution:

The bases are the same and m > n. Thus applying the quotient law, 6⁹ ÷ 6³ = 6⁹⁻³ = 6⁶.

Express 𝟏𝟕^𝟏𝟐 ÷ 𝟏𝟕^𝟐𝟏in exponential form. Solution:

The bases are the same, but m < n. Thus, applying the variation of the quotient law, 17¹² ÷

17²¹ = 1/1721−12=1/17⁹

Summary

Quotient Law of Exponents	For any non-zero integer a, 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛, where m and n are whole numbers and m > n.
Variation of Quotient Law	For any non-zero integer a, 𝑎𝑚 ÷ 𝑎𝑛 = 1 , where m and n 𝑎𝑛−𝑚 are whole numbers and m < n.