Multiplying, dividing and raising monomials to powers
The exponent rules you need for monomials, and how to multiply, divide and raise monomials to a power, with worked examples and common mistakes.
The exponent rules
Everything in this lesson rests on a few rules for exponents with the same base. They all come from counting factors.
For instance, \(x^2 \cdot x^3 = (x \cdot x)(x \cdot x \cdot x) = x^5\): two factors and three factors make five. And \(\left(x^2\right)^3 = x^2 \cdot x^2 \cdot x^2 = x^6\): three groups of two.
Multiplying monomials
Unlike addition, multiplication works for any two monomials, alike or not.
The degree of a product is the sum of the degrees. In the first example, a degree-4 monomial times a degree-5 monomial gives degree 9.
Dividing monomials
Raising a monomial to a power
Putting it together
Expressions with monomials follow the usual order of operations: powers first, then multiplication and division from left to right, then addition and subtraction.
Practice
Quick check
What is \((5a^4)(2a^3)\)?
Multiply the coefficients (\(5 \cdot 2 = 10\)) and add the exponents of \(a\) (\(4 + 3 = 7\)).
Quick check
What is \(\left(-y^2\right)^3\)?
An odd power of a negative is negative, and the exponents multiply: \(2 \cdot 3 = 6\).
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