Basic Technical Mathematics, 11th Edition

11.3 Simplest Radical Form

• Operations with Radicals • Removing Perfect nth-power Factors • Reducing the Order of a Radical • Rationalizing the Denominator.

As we have said, any expression with radicals can also be expressed with fractional exponents. For adding or subtracting radicals, there is little advantage to changing form, but with multiplication or division of radicals, fractional exponents have some advantages. Therefore, we now define operations with radicals so that they are consistent with the laws of exponents. This will let us use the form that is more convenient for the operation being performed.

$$\sqrt[n]{a^n}\hspace{0.17em}=\hspace{0.17em}{(\sqrt[n]{a})}^n\hspace{0.17em}=\hspace{0.17em}a$$(11.9)

$$\sqrt[n]{a}\sqrt[n]{b}\hspace{0.17em}=\hspace{0.17em}\sqrt[n]{ab}$$(11.10)

$$\sqrt[m]{\sqrt[n]{a}}\hspace{0.17em}=\hspace{0.17em}\sqrt[mn]{a}$$(11.11)

$${\displaystyle \frac{\sqrt[n]{a}}{\sqrt[n]{b}}}\hspace{0.17em}=\hspace{0.17em}\sqrt[n]{{\displaystyle \frac{a}{b}}}(b\hspace{0.17em}\ne \hspace{0.17em}0)$$(11.12)

The number under the radical is called the radicand, ...