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Evaluate the Square Root of (11⁴+100⁴+111⁴)/2 | IOQM

JEE Maths question with a full step-by-step solution.

Question

Calculate the value of

\sqrt{\dfrac{11^4 + 100^4 + 111^4}{2}}

Solution

Answer: 11221

Step 1: Key identity. For all real

a, b

a^4 + b^4 + (a + b)^4 = \frac{1}{2}\big[a^2 + b^2 + (a + b)^2\big]^2.

Step 2: Since

111 = 11 + 100

, take

a = 11

b = 100

a + b = 111

. Then

11^4 + 100^4 + 111^4 = \frac{1}{2}\big[11^2 + 100^2 + 111^2\big]^2,

\frac{11^4 + 100^4 + 111^4}{2} = \frac{1}{4}\big[11^2 + 100^2 + 111^2\big]^2.

Step 3: Take the square root.

\sqrt{\frac{11^4 + 100^4 + 111^4}{2}} = \frac{11^2 + 100^2 + 111^2}{2} = \frac{121 + 10000 + 12321}{2} = \frac{22442}{2} = 11221.

Answer:

11221

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