Find the square root of the following surds
12+2root 35
8-2root 15
31-4root 21
30+12root 6

1 thought on “Find the square root of the following surds<br /> 12+2root 35<br /> 8-2root 15<br /> 31-4root 21<br /> 30+12root 6”

1. Given:

   [tex]\bf 1. \: 12+2\sqrt{35}[/tex]

   [tex]\bf 2. \: 8-2\sqrt{15}[/tex]

   [tex]\bf 3. \: 31-4\sqrt{21}[/tex]

   [tex]\bf 4. \: 30+12\sqrt{6}[/tex]

   What To Find:

   We have to find –

   • The square root of the given surds.

   How To Find:

   To find, we have to –

   • First, write in the form of [tex]\sf \sqrt{a}+\sqrt{b}[/tex].
   • Next. square the sides by using identities.
   • Then, find the square roots of the expression.

   Answer 1:

   [tex]\sf 12+2\sqrt{35}[/tex]

   Take it in the form of [tex]\sf \sqrt{a}+\sqrt{b}[/tex] and write it as,

   [tex]\sf \to \sqrt{a}+\sqrt{b} = \sqrt{12+2\sqrt{35}}[/tex]

   Square both sides,

   [tex]\sf \to (\sqrt{a}+\sqrt{b})^2 = \bigg(\sqrt{12+2\sqrt{35}}\bigg)^2[/tex]

   Square the LHS by using the identity,

   [tex]\sf \to (\sqrt{a}+\sqrt{b})^2 = a + b + 2\sqrt{ab}[/tex]

   Square the RHS by cancelling the surds and powers,

   [tex]\sf \to\bigg(\sqrt{12+2\sqrt{35}}\bigg)^2 = 12+2\sqrt{35}[/tex]

   After squaring both sides,

   [tex]\sf \to a + b + 2\sqrt{ab} = 12 + 2\sqrt{35}[/tex]

   Let’s take –

   • a + b = 12
   • 2√ab = 2√35

   Cancel 2 from both sides,

   • √ab = √35

   On squaring,

   • (√ab)² = (√35)²
   • ab – 35

   By manual guess,

   • a + b = 12 = 7 + 5
   • ab = 35 = 7 × 5

   ∴ Thus, the answer is:- [tex]\bf \sqrt{7}+\sqrt{5}[/tex].

   Answer 2:

   [tex]\sf 8-2\sqrt{15}[/tex]

   Take it in the form of [tex]\sf \sqrt{a}+\sqrt{b}[/tex] and write it as,

   [tex]\sf \to \sqrt{a}+\sqrt{b} = \sqrt{8-2\sqrt{15}}[/tex]

   Square both sides,

   [tex]\sf \to (\sqrt{a}+\sqrt{b})^2 = \bigg(\sqrt{8-2\sqrt{15}}\bigg)^2[/tex]

   Square the LHS by using the identity,

   [tex]\sf \to (\sqrt{a}+\sqrt{b})^2 = a + b + 2\sqrt{ab}[/tex]

   Square the RHS by cancelling the surds and powers,

   [tex]\sf \to\bigg(\sqrt{8-2\sqrt{15}}\bigg)^2 = 8-2\sqrt{15}[/tex]

   After squaring both sides,

   [tex]\sf \to a + b + 2\sqrt{ab} = 8-2\sqrt{15}[/tex]

   Let’s take –

   • a + b = 8
   • 2√ab = 2√15

   Cancel 2 from both sides,

   • √ab = √15

   On squaring,

   • (√ab)² = (√15)
   • ab = 15

   By manual guess,

   • a + b = 8 = 5 + 3
   • ab = 15 = 5 × 3

   ∴ Thus, the answer is:- [tex]\bf \sqrt{5} + \sqrt{3}[/tex]

   Answer 3:

   [tex]\sf 31-4\sqrt{21}[/tex]

   Take it in the form of [tex]\sf \sqrt{a}+\sqrt{b}[/tex] and write it as,

   [tex]\sf \to \sqrt{a}+\sqrt{b} = \sqrt{31-4\sqrt{21}}[/tex]

   Square both sides,

   [tex]\sf \to (\sqrt{a}+\sqrt{b})^2 = \bigg(\sqrt{31-4\sqrt{21}}\bigg)^2[/tex]

   Square the LHS by using the identity,

   [tex]\sf \to (\sqrt{a}+\sqrt{b})^2 = a + b + 2\sqrt{ab}[/tex]

   Square the RHS by cancelling the surds and powers,

   [tex]\sf \to\bigg(\sqrt{31-4\sqrt{21}}\bigg)^2 = 31-4\sqrt{21}[/tex]

   After squaring both sides,

   [tex]\sf \to a + b + 2\sqrt{ab} = 31-4\sqrt{21}[/tex]

   Let’s take –

   • a + b = 31
   • 2√ab = 4√21

   Cancel 2 from both sides,

   • √ab = 2√21

   Squaring both sides,

   • (ab)² = (2√21)²
   • ab = 84

   By manual guess,

   • a + b = 31 = 28 + 3
   • ab = 84 = 28 × 3

   ∴ Thus, the answer is:- [tex]\bf \sqrt{28}+\sqrt{3}[/tex] or [tex]\bf 2\sqrt{7} + \sqrt{3}[/tex]

   Answer 4:

   [tex]\sf 30+12\sqrt{6}[/tex]

   Take it in the form of [tex]\sf \sqrt{a}+\sqrt{b}[/tex] and write it as,

   [tex]\sf \to \sqrt{a}+\sqrt{b} = \sqrt{30+12\sqrt{6}}[/tex]

   Square both sides,

   [tex]\sf \to (\sqrt{a}+\sqrt{b})^2 = \bigg(\sqrt{30+2\sqrt{6}}\bigg)^2[/tex]

   Square the LHS by using the identity,

   [tex]\sf \to (\sqrt{a}+\sqrt{b})^2 = a + b + 2\sqrt{ab}[/tex]

   Square the RHS by cancelling the surds and powers,

   [tex]\sf \to\bigg(\sqrt{30+12\sqrt{6}}\bigg)^2 = 30+12\sqrt{6}[/tex]

   After squaring both sides,

   [tex]\sf \to a + b + 2\sqrt{ab} = 30+12\sqrt{6}[/tex]

   Let’s take –

   • a + b = 30
   • 2√ab = 12√6

   Cancel 2 from both sides,

   • ab = 6√6

   Squaring both sides,

   • (√ab)² = (6√6)²
   • ab = 216

   By manual guess,

   • a + b = 30 = 12 + 18
   • ab = 216 = 28 × 3

   ∴ Thus, the answer is:- [tex]\bf \sqrt{18} + \sqrt{12}[/tex] or [tex]\bf 3\sqrt{2} + 2\sqrt{3}[/tex]

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