Question

Compare $\sqrt[3]{364},\sqrt[3]{380}$.

• A. $\sqrt[3]{364}>\sqrt[3]{380}$
• B. $\sqrt[3]{364}<\sqrt[3]{380}$
• C. $\sqrt[3]{364}=\sqrt[3]{380}$
• D. None of the above

Answer 1

The correct option is B $\sqrt[3]{364}<\sqrt[3]{380}$

We have to find the approximate values of $\sqrt[3]{364}$ and $\sqrt[3]{380}$ by perfect cube method.
As we know $\sqrt[3]{364}=\sqrt[3]{34\times 4\times 4}=4$

Now, let's find approximate value of $\sqrt[3]{380}$,
Largest perfect cube less than $80$ is $64$ whose cube root is $4$
Smallest perfect cube greater than $80$ is $125$ whose cube root is $5$.
$\Rightarrow 64<80<125\Rightarrow 4<\sqrt[3]{380}<5$

Since $\sqrt[3]{380}>4$ and $\sqrt[3]{364}=4$. we can say that $\sqrt[3]{364}<\sqrt[3]{380}$.

Alternatively, we can compare both $\sqrt[3]{364}\&\sqrt[3]{380}$ by comparing cube of both.
$(\sqrt[3]{364}{)}^3=64\&(\sqrt[3]{380}{)}^3=80$
As, taking cube of a number does not affect its positive or negative sign.
$\because 80>64\Rightarrow (\sqrt[3]{380}{)}^3>(\sqrt[3]{364}{)}^3$

If $a^3>b^3\Rightarrow a>b$

$\therefore \sqrt[3]{380}>\sqrt[3]{364}$