Question

Simplify: \[3\sqrt[3]{40}-4\sqrt[3]{320}-\sqrt[3]{5}\].

Answer 1

Hint: In order to simplify the given terms, firstly we have to expand the first and the second terms into the products and express them in a simpler form. After expressing in simpler form, we have to find the common terms from all of the three simplified terms and then we have to evaluate them. This would give us the required answer.

Complete step by step answer:
Now let us have brief regarding the cube roots. The cube root of a number is nothing but when a number is multiplied thrice, we obtain the cube of the number. The number being multiplied is called the cube root of the obtained cube number. All non-zero real numbers have exactly one cube root . It also has one pair of conjugate cube roots. All of the non zero complex numbers have three distinct complex cube roots.
Now let us simplify the given terms: \[3\sqrt[3]{40}-4\sqrt[3]{320}-\sqrt[3]{5}\]
Now let us simplify the first term into simplest form.
\[3\sqrt[3]{40}=3\sqrt[3]{8\times 5}\]
This can be expressed by splitting in the following way-
\[3\sqrt[3]{8\times 5}=3\left( \sqrt[3]{8}\times \sqrt[3]{5} \right)\]
\[\because \sqrt[3]{8}=2\], we get
\[\Rightarrow 3\times 2\left( \sqrt[3]{5} \right)=6\sqrt[3]{5}\]
Now, the simplest form of second term would be-
\[4\sqrt[3]{320}=4\sqrt[3]{5\times 8\times 8}\]
Upon splitting the terms we get,
\[\Rightarrow 4\sqrt[3]{5\times 8\times 8}=4\left( \sqrt[3]{5}\times \sqrt[3]{8}\times \sqrt[3]{8} \right)\]
\[\because \sqrt[3]{8}=2\], we get
\[\Rightarrow 4\times 2\times 2\left( \sqrt[3]{5} \right)=16\sqrt[3]{5}\]

We can see that the third term is already in its simplest form and i.e. \[\sqrt[3]{5}\].
Now let us substitute the simplest terms in place of the original terms. We get
\[\Rightarrow 6\sqrt[3]{5}-16\sqrt[3]{5}-\sqrt[3]{5}\]
We can see that \[\sqrt[3]{5}\] is the common term in all of the terms. So let us take \[\sqrt[3]{5}\] as common and then simplify it.
\[\begin{align}
  & \Rightarrow \sqrt[3]{5}\left( 6-16-1 \right) \\
 & \Rightarrow \sqrt[3]{5}\left( -11 \right) \\
 & \Rightarrow -11\sqrt[3]{5} \\
\end{align}\]
\[\therefore \] \[3\sqrt[3]{40}-4\sqrt[3]{320}-\sqrt[3]{5}\]\[=-11\sqrt[3]{5}\]

Note: We must always have a note that if the terms are not in the simplest terms or made common, we cannot simplify them. So it is important for us to make them common and consider the common terms for easier simplification.