Radical simplification

Product and Quotient rule

Let [math]n[/math] be a positive integer number, [math]n\ge2[/math] and let [math]a[/math] and [math]b[/math] be two real numbers, such that [math]\sqrt[n]{a}[/math] and [math]\sqrt[n]{b}[/math] are well defined. Then:[br][br][list][*][math]\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{a\cdot b}[/math].[/*][*][math]\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}[/math]; provided that [math]b\ne0[/math].[/*][/list]

Simplified form

[b]Square root:[/b][br]A square root radical is in simplified form if it complies the following three conditions:[br]1. The number under the radical (radicand) has no factor, but 1, that is perfect square.[br]2. Radicand has no fractions.[br]3. No denominator has radicals.[br][br][math]n-th[/math] [b]root:[/b][br]An [math]n-th[/math] root radical is in simplified form if the following three conditions are met:[br]1. The number under the radical (radicand) has no factor, but 1, that is an [math]n-th[/math] power.[br]2. Radicand has no fractions.[br]3. No denominator has radicals

Guidelines to simplify a radical

Square root:[br][list=1][*]Factor out the coefficient as the product of the highest perfect square that divides it and a number that is not a perfect square.[/*][*]Write the variables as a product of the highest even power less than the exponent, and an odd power.[/*][*]Take square root of the perfect squares.[/*][/list][br][math]n-th[/math] [b]root:[/b][br][list=1][*]Factor out the coefficient as the product of the highest [math]n-th[/math] power that divides it and a number that is not an [math]n-th[/math] power.[/*][*]Write the variables as a product of the highest [math]n-th[/math] power less than the exponent, and the difference between the exponent and the [math]n-th[/math] power.[/*][*]Take [math]n-th[/math] root of the [math]n-th[/math] powers.[/*][/list]

Formative assessment

Use the following applets to practice simplifying radicals.

References

Brzezinski, T. (2016, September 2). [i]Simplifying radicals (i)[/i]. GeoGebra. https://www.geogebra.org/m/g7UAQHDK. [br][br][br]Brzezinski, T. (2016, September 2). [i]Simplifying radicals (ii)[/i]. GeoGebra. https://www.geogebra.org/m/nufHUrek. [br][br][br]Miller, C. D., Heeren, V. E., Hornsby, J., & Heeren, C. (2020). [i]Mathematical ideas[/i]. Pearson Education, Inc. [br][br]