UPSKILL MATH PLUS

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2. Cube root through prime factorisation

Steps to find the cube root of a number through prime factorisation:

Step 1: Find the prime factorisation of the given number.

Step 2: Group the factors in pair of three numbers (triplet).

Step 3: If there are no factor leftover, then the given number is a perfect cube. Otherwise, it is not a perfect cube.

Step 4: Now, take one factor common from each pair and multiply them.

Step 5: The obtained product is a cube root of a given number.

Example:

1. Find the value of \(\sqrt[3]{216}\).

Solution:

Let us first find the prime factor of \(216\).

\begin{array}{l} \underline{2 | 216} \\ \underline{2 | 108} \\ \underline{2 | 54} \\ \underline{3 | 27} \\ \underline{3 | 9} \\ \underline{3 | 3} \\ | 1 \end{array}

Group the factors in pair of three numbers.

\(216 = (2 \times 2 \times 2) \times (3 \times 3 \times 3)\)

Here, no factor is leftover. Therefore, \(216\) is a perfect cube.

Now, take one factor common from each pair and multiply them.

\(\sqrt[3]{216} = 2 \times 3 = 6\)

Therefore, the value of \(\sqrt[3]{216} = 6\).

2. Evaluate the value of

\sqrt[3]{\frac{2197}{1000}}

Solution:

The given number is

\sqrt[3]{\frac{2197}{1000}}

Let us first factorise the numbers \(2197\) and \(1000\).

\sqrt[3]{\frac{2197}{1000}}

\(=\)

\frac{\sqrt[3]{2197}}{\sqrt[3]{1000}}

\(=\)

\frac{\sqrt[3]{13 \times 13 \times 13}}{\sqrt[3]{10 \times 10 \times 10}}

\(=\)

\frac{\sqrt[3]{13^{3}}}{\sqrt[3]{10^{3}}}

\(=\)

\frac{13}{10}

(Cube and cube root get cancelled)

Therefore, the value of

\sqrt[3]{\frac{2197}{1000}}

\(=\)

\frac{13}{10}

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