Square roots and Cube roots

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Square Roots

The square root of a positive number ‘n’ is that number which, when multiplied by itself, gives ‘n’.

There are two square roots for a positive number. For example, the square roots of 25 are +5 and -5. The positive square root of a number is called the principal square root.

The symbol √ is called the radical sign. The index notation is also used instead of the radical sign. For example:

\sqrt{16} = +4 can be written as  16^{1/2} = +4 .

square roots and cube roots

 

Example: Find the square root of 81.

81 = 9 × 9

\sqrt{81}= 9

Cube Roots

The cube root of a positive number ‘N’ is that number which, when multiplied by itself twice, gives ‘N’.

For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8 (here 2 is being multiplied by itself twice)

The symbol for cube root is a radical sign with a small 3 on the left ∛ .

Example: Find the cube root of -125.

-125 = (-5)3 = -5 × -5 × -5

\sqrt[3]{-125} = -5

Example: Simplify: \sqrt[3]{343}  – \sqrt{196} + 4

7 × 7 × 7 = 343, so  \sqrt[3]{343}  = 7

14 × 14 = 196, so \sqrt{196} = 14

so , \sqrt[3]{343}  – \sqrt{196} + 4 = 7 – 14 + 4 = 11 -14 = -3

CHECK POINT 

  1. Find the square root of 169.
  2. What is the cube root of 1331?
  3. Simplify: \sqrt{225} + \sqrt[3]{-729} – 5
  4. Value of \sqrt[3]{216} – \sqrt[3]{-216}
  5. What is the square root of 1,024?
Answer key
  1. 13
  2. 11
  3. 1
  4. 12
  5. 32