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If sqrt32+sqrt8=asqrt2, then what is the value of a ?

If 32+8=a2 \sqrt{32}+\sqrt{8}=a \sqrt{2} , then what is the value of a a ?

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Full solution

Q. If 32+8=a2 \sqrt{32}+\sqrt{8}=a \sqrt{2} , then what is the value of a a ?
  1. Simplify 32\sqrt{32}: Simplify 32\sqrt{32} and 8\sqrt{8} by expressing them as multiples of 2\sqrt{2}.\newline32\sqrt{32} can be written as 16×2\sqrt{16 \times 2} which simplifies to 424\sqrt{2}.\newline8\sqrt{8} can be written as 4×2\sqrt{4 \times 2} which simplifies to 222\sqrt{2}.
  2. Add simplified forms: Add the simplified forms of 32\sqrt{32} and 8\sqrt{8} together.\newline42+22=(4+2)2=62.4\sqrt{2} + 2\sqrt{2} = (4 + 2)\sqrt{2} = 6\sqrt{2}.
  3. Compare with given equation: Compare the result from Step 22 with the given equation 32+8=a2\sqrt{32} + \sqrt{8} = a\sqrt{2}. We have 62=a26\sqrt{2} = a\sqrt{2}.
  4. Solve for value of aa: Solve for the value of aa.\newlineSince 62=a26\sqrt{2} = a\sqrt{2}, we can equate the coefficients of 2\sqrt{2} on both sides of the equation to find the value of aa.\newlineTherefore, a=6a = 6.

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