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Skip to 0 minutes and 10 secondsHello. We have to find the real numbers, a, such that this expression makes sense, and simplify it. OK. Remember that roots are defined only when the argument is a positive number. But since a to the 4th is greater or equal than 0 for any real number a, then our expression is defined--

Skip to 0 minutes and 57 secondsis defined for all real numbers.

Skip to 1 minute and 6 secondsOK, let us try to simplify it. OK, let us write our expression--

Skip to 1 minute and 16 secondsin this form, first-- it's clear that 8 is 2 times 4. a to the 4th is exactly like to write, the module, of a to the 4th. And now we have the square. OK. Now we can simplify the expression inside, between brackets. And we get the square root of the module of a, and then to the square. And this now is equal to module of a.

Skip to 2 minutes and 16 secondsWe want to find all the real numbers for which this expression makes sense, and then try to simplify it. OK, this expression can be rewritten in the following way--

Skip to 2 minutes and 35 secondslike the 6th root of a to 9 over the 3rd root of a. And now you know that roots are defined only for real numbers greater or equal than 0. But now we have also something in the denominator. And this has to be not equal to 0. Therefore, this expression is defined--

Skip to 3 minutes and 9 secondsfor all the real numbers a, greater than 0. And now, under this assumption, let us try to simplify this fraction. OK, we can write again this expression in the following form-- as-- the 6th root of a to the 9. And here we can write in this following form-- the 6th root of a squared is exactly the same. And now we have the same order of roots in the numerator and the denominator. And we get the 6th root of a to the 9 over a to the square, which is equal to the 6th root of a to the 9 minus 2-- a to the 7th power.

Roots and radicals in practice

The following exercises are solved in this step.

We invite you to try to solve them before watching the video.

In any case, you will find below a PDF file with the solutions.

Exercise 1.

Find all the values of \(a \in \mathbb R\) for which the following expression makes sense and then simplify it: \[\left(\sqrt[8]{a^4}\right)^2\]

Exercise 2.

Find all the values of \(a \in \mathbb R\) for which the following expression makes sense and then simplify it: \[\left(\sqrt[3]{a}\right)^{-1}\sqrt[6]{a^9}\]

Ex 3. [Solved only in the PDF file]

Sketch the graph of the function \(f:x\mapsto\sqrt[3]x\).

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Precalculus: the Mathematics of Numbers, Functions and Equations

University of Padova

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