Current Affairs 8th Class

Category : 8th Class

 Cubes and Cube Roots

• Cubes

(i) The cube of a number is the product of the number multiplied by itself twice.

(ii) Write the cube of a number using the cube symbol or notation.

(iii) \[{{8}^{3}}\]is read as 'eight cubed' or 'the cube of eight', or 'eight to the power of three.

• Estimating the cubes of numbers

Estimate the cube of a number by determining the range in which its value lies. e.g. Estimate the cube of 10.6 by determining the range in which its value lies.

Solution

10 < 10.6 < 11 \[\leftarrow \] Determine the range

\[103<{{\left( 10.6 \right)}^{3}}<113\]\[\leftarrow \]Cube the range

\[1000<{{\left( 10.6 \right)}^{3}}<1331\]Estimated answer

\[\therefore {{\left( 10.6 \right)}^{3}}\]is between 1000 and 1331.

• Perfect cube

(i) A natural number is said to be a perfect cube if it is the cube of some natural number.

(ii) Cubes of all even natural numbers are even.

(iii) Cubes of all odd natural numbers are odd.

(iv) Cubes of negative integers are negative.

• Cube roots of numbers

(i) The cube root of a number is a number which, when multiplied by itself twice, equals the given number.

(ii) The symbol used for cube root is\[\sqrt{3}\]

(iii) The cube root of a number  \['x'\] is that number whose cube gives\['x'\]. It is denoted as  

(iv) For any positive integer \['x',\sqrt[3]{-x}=-\sqrt[3]{x}\]

For any two integers 'a' and \['b'\]

(a) \[\sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}\]     (b)\[\sqrt[3]{\frac{a}{b}}=\frac{\sqrt{3}}{\sqrt{3}}\]

(vi) Cube root of a number can be found by prime fatorisation.

Determining the cube roots

(i) To find the cube roots of fractions, reduce the fractions to numerators and denominators that are cubes of integers. Then, find the cube roots of those integers.

(ii) The find the cube roots of decimals, convert the decimals to fractions so that the numerators and denominators are cubes of integers. Then, solve the cube roots of those integers.