Converting Radicals to Rational Exponents
A power with a rational exponent m/n of a real number a≥0 is equal to the n-th root of am $$ a^{\frac{m}{n}}=\sqrt[n]{a^m}$$
This relationship allows us to apply the laws of exponents even in expressions that contain radicals.
In many cases, rewriting radicals as powers makes algebraic calculations easier and more systematic.
Note. In the special case where m<0, the base must be strictly positive, a>0. If a=0, the expression would involve division by zero, which is undefined. $$ a^{-m} = \frac{1}{a^m}$$
A practical example
Consider the following power with a rational exponent
$$ 5^{\frac{3}{4}} $$
We can rewrite this expression as a radical whose index is 4 and whose radicand is raised to the power 3
$$ 5^{\frac{3}{4}} = \sqrt[4]{5^3} $$
This form is completely equivalent to the original expression.
Example 2
Consider the following radical
$$ \sqrt[6]{a^3} $$
This radical can be simplified. Instead of applying the usual rules for simplifying radicals, we first rewrite it as a power with a rational exponent.
$$ \sqrt[6]{a^3} = a^{\frac{3}{6}} $$
Next, simplify the fraction 3/6 in the exponent
$$ \sqrt[6]{a^3} = a^{\frac{1}{2}} $$
Finally, convert the power back into radical form
$$ \sqrt[6]{a^3} = a^{\frac{1}{2}} = \sqrt{a} $$
In this way the radical is simplified by applying the laws of exponents.
Note. The same result can also be obtained using the simplification rule for radicals, dividing both the index and the exponent of the radicand by their greatest common divisor. $$ \sqrt[6]{a^3} = \sqrt[\frac{6}{3}]{a^{\frac{3}{3}}} = \sqrt{a}$$ The final result is the same.
Example 3
Consider the following sum of radicals
$$ 2\sqrt[4]{a^3} + 3 \sqrt[4]{a^3} $$
First rewrite the radicals as powers with rational exponents
$$ 2 a^{\frac{3}{4}} + 3 a^{\frac{3}{4}} $$
Then factor out the common power and add the coefficients
$$ (2+3)\cdot a^{\frac{3}{4}} $$
$$ 5 a^{\frac{3}{4}} $$
Finally, convert the power with a rational exponent back into radical form
$$ 5 a^{\frac{3}{4}} = 5 \sqrt[4]{a^3} $$
This is exactly the same result that would be obtained by directly adding the radicals.
Check. The radicals have the same index and the same radicand, so their coefficients can be added. $$ 2\sqrt[4]{a^3} + 3 \sqrt[4]{a^3} = (2+3)\cdot \sqrt[4]{a^3} = 5 \sqrt[4]{a^3} $$ The result is the same.
The same reasoning can be applied to many similar expressions.
