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sqrt128+sqrt8

128+8 \sqrt{128}+\sqrt{8}

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Full solution

Q. 128+8 \sqrt{128}+\sqrt{8}
  1. Identify Prime Factors: Identify the prime factors of 128128 and 88 to simplify the square roots.\newline128=2×64=2×2×32=2×2×2×16=2×2×2×2×8=2×2×2×2×2×4=27128 = 2 \times 64 = 2 \times 2 \times 32 = 2 \times 2 \times 2 \times 16 = 2 \times 2 \times 2 \times 2 \times 8 = 2 \times 2 \times 2 \times 2 \times 2 \times 4 = 2^7\newline8=2×4=2×2×2=238 = 2 \times 4 = 2 \times 2 \times 2 = 2^3
  2. Break Down Square Roots: Break down the square roots using the prime factors.\newline128=27=26×2=23×2=82\sqrt{128} = \sqrt{2^7} = \sqrt{2^6} \times \sqrt{2} = 2^3 \times \sqrt{2} = 8\sqrt{2}\newline8=23=22×2=22\sqrt{8} = \sqrt{2^3} = \sqrt{2^2} \times \sqrt{2} = 2\sqrt{2}
  3. Add Simplified Square Roots: Add the simplified square roots together.\newline82+22=(8+2)2=1028\sqrt{2} + 2\sqrt{2} = (8 + 2)\sqrt{2} = 10\sqrt{2}

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