Question

Using cube root table, find the cube root of 309400.

Answer 1

The cubed root of three hundred and nine thousand, four hundred ∛309400 = 67.6353024531

Answer 2

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Solution :

The cube root table gives cube roots of natural numbers upto 9900.
Clearly, 309400 is greater than 9900.

So, we write

309400= 1547 × 200


$\sqrt[3]{309400} = \sqrt[3]{1547 \times 200} \\ \\ = \sqrt[3]{1547} \times \sqrt[3]{200}$


Now, 1500 < 1547 < 1600

$= \sqrt[3]{1500} < \sqrt[3]{1547} < \sqrt[3]{1600}$


From the cube root table, we have

$\sqrt[ 3]{1500} = 11.45 \: \: and \: \: \sqrt[3]{1600} = 11.70$


Thus, for the difference of ( 1600 - 1500 ) i.e. 100, we have,
The difference in the values = 11.70 - 11.45 = 0.25

For the difference of ( 1547 - 1500 ) i.e 47, we have

The difference in the values =
$\frac{0.25}{100} \times 47 \\ \\ = 0.1175 = 0.117 \: \: \: \: ( \: upto \: \: three \: \: decimal \: \: places) \\ \\ = \sqrt[3]{1547} = 11.45 + 0.117 = 11.567$


Also, from the cube root table, we have

$\sqrt[3]{200} = 5.848 \\ \\ = \sqrt[3]{309400} = \sqrt[3]{1547} \times \sqrt[ 3 ]{200} = 11.567 \times 5.848 = 67.643$