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Упростите выражение (sqrt(sqrt10-2)*sqrt(sqrt10+2))/(sqrt24).

Упростите выражение 10−2⋅10+224 \frac{\sqrt{\sqrt{10}-2} \cdot \sqrt{\sqrt{10}+2}}{\sqrt{24}} .

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Full solution

Q. Упростите выражение 10−2⋅10+224 \frac{\sqrt{\sqrt{10}-2} \cdot \sqrt{\sqrt{10}+2}}{\sqrt{24}} .
  1. Recognize Structure: Recognize the structure of the numerator as a difference of squares. The expression inside the square roots in the numerator resembles the difference of squares formula: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b). Here, a=10a = \sqrt{10} and b=2b = 2.
  2. Apply Formula: Apply the difference of squares formula to the numerator.\newline(10−2)(10+2)=(10)2−(2)2=10−4=6.(\sqrt{10} - 2)(\sqrt{10} + 2) = (\sqrt{10})^2 - (2)^2 = 10 - 4 = 6.\newlineSo, the numerator simplifies to 6\sqrt{6}.
  3. Simplify Denominator: Simplify the denominator.\newlineThe denominator is 24\sqrt{24}. We can simplify this by factoring out perfect squares.\newline24=4×6=4×6=2×6\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2 \times \sqrt{6}.
  4. Divide Numerator: Divide the simplified numerator by the simplified denominator.\newlineNow we have (6)/(2⋅6)(\sqrt{6}) / (2 \cdot \sqrt{6}).\newlineWe can simplify this by dividing 6\sqrt{6} by 6\sqrt{6}, which is 11, and then by 22.\newline(6)/(2⋅6)=1/2(\sqrt{6}) / (2 \cdot \sqrt{6}) = 1 / 2.

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