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B. Multiplying and Dividing Radicals: Monomials


 Warm Up

Review of the Exponent Laws


The exponent laws need to be reviewed in order to understand and perform multiplication and division with radicals.



  Key Lesson Marker

Exponent Laws


The Exponent Laws are:

Product Law: Multiplication of “like” bases: add the exponents.

\(x^m x^n = x^{m + n} \)

Division Law: Division of “like” bases: subtract the exponents.
\[\frac{{x^m}}{{x^n}} = x^{m - n}, \thinspace x \ne 0\]

Power of a Power Law: Multiply the exponents.

\(\left( {x^m} \right)^n = x^{m\cdot n} \)

Power of a Product Law: Distribute the exponent to all factors.

\(\left( {xy} \right)^m = x^m y^m\)

Power of a Quotient Law: Distribute the exponent to the dividend and the divisor.
\[\left( {\frac{x}{y}} \right)^m = \frac{{x^m }}{{y^m }}, \thinspace y \ne 0\]

Zero Exponent Law: Any value to the exponent zero is equal to \(1\).

\(x^0 = 1, \thinspace x \ne 0\)

Negative Exponent Law: Power with a negative exponent is equal to its reciprocal with a positive exponent.
\[x^{-m} = \frac{1}{{x^m }}, \thinspace x \ne 0\]

Rational Exponents Law: The denominator of the exponent becomes the index of the radical, and the numerator becomes the exponent of the power in the radicand.
\[x^{\frac{m}{n}} = \sqrt[n]{{x^m }}\]