簡約 sqrt(13+4\sqrt{3)}

解

簡約

13 + 43

23 + 1

十進法表記

4.46410 \dots

解答ステップ

13 + 43

= 12 + 43 + 1

= 4 \cdot 3 + 43 + 1

= (4)^{2} (3)^{2} + 43 + (1)^{2}

4 = 2

1 = 1

= 2^{2} (3)^{2} + 43 + 1^{2}

2 \cdot 23 \cdot 1 = 43

= (23)^{2} + 2 \cdot 23 \cdot 1 + 1^{2}

完全平方式を適用する:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(23)^{2} + 2 \cdot 23 \cdot 1 + 1^{2} = (23 + 1)^{2}

= (23 + 1)^{2}

累乗根の規則を適用する:

n a^{n} = a

(23 + 1)^{2} = 23 + 1

= 23 + 1