Penjelasan dengan langkah-langkah: [tex]\displaystyle(i.)\:\:\because\sqrt{p+q+\sqrt{4pq}}=\sqrt{p}+\sqrt{q}\therefore\\\\\sqrt{21+4\sqrt{20}}=\sqrt{21+\sqrt{4^2}\sqrt{20}}\\\\=\sqrt{21+\sqrt{4(4(20))}}=\sqrt{21+\sqrt{4(80)}}\\\\p+q=21,\:\:\:pq=80\\\\Vieta.\:\:\{A=1\}\\p+q=-B,\:\:pq=C\\maka\:\:-B=21,\:\:C=80\\\\Bentuk\:\:\:persamaan\:\:\:kuadrat\\Ax^2+Bx=-C\\x^2-21x=-80\\x^2-2\left(\frac{21}{2}\right)x+\left(\frac{21}{2}\right)^2=-80+\left(\frac{21}{2}\right)^2\\\because u^2-2uv+v^2=(u-v)^2\therefore[/tex] [tex]\displaystyle\left(x-\frac{21}{2}\right)^2=-80+\left(\frac{21}{2}\right)^2\\\\\left(x-\frac{21}{2}\right)^2=-\frac{320}{4}+\frac{441}{4}\\\\\left(x-\frac{21}{2}\right)^2=\frac{121}{4}\\\\\left|x-\frac{21}{2}\right|=\sqrt{\frac{121}{4}}\\\\\left|x-\frac{21}{2}\right|=\frac{\sqrt{121}}{\sqrt{4}}\\\\\left|x-\frac{21}{2}\right|=\frac{11}{2}\\\\x=\left\{\frac{11}{2}+\frac{21}{2},\:\:\:\:-\frac{11}{2}+\frac{21}{2}\right\}\\\\p=\frac{11}{2}+\frac{21}{2},\:\:\:q=-\frac{11}{2}+\frac{21}{2}[/tex] [tex]\displaystyle p=\frac{32}{2},\:\:\:q=\frac{10}{2}[/tex] p = 16, q = 5 Oleh karena itu hasil sederhananya [tex]\sqrt{p}+\sqrt{q}=\sqrt{16}+\sqrt{5}=\\\bf4+\sqrt{5}[/tex]
Jawab:
[tex]\rm(i)\:\:\:\:\sqrt{21+4\sqrt{20}}\:\:\:\bf=\:\:\:4+\sqrt{5}\\\rm(ii)\:\:\:\sqrt{75} -2\sqrt{48} -\sqrt{147} +3\sqrt{27} =-\sqrt{3}[/tex]
(iii) (8,6×10⁸) × (4×10⁶) = 3,44 × 10¹⁵
Penjelasan dengan langkah-langkah:
[tex]\displaystyle(i.)\:\:\because\sqrt{p+q+\sqrt{4pq}}=\sqrt{p}+\sqrt{q}\therefore\\\\\sqrt{21+4\sqrt{20}}=\sqrt{21+\sqrt{4^2}\sqrt{20}}\\\\=\sqrt{21+\sqrt{4(4(20))}}=\sqrt{21+\sqrt{4(80)}}\\\\p+q=21,\:\:\:pq=80\\\\Vieta.\:\:\{A=1\}\\p+q=-B,\:\:pq=C\\maka\:\:-B=21,\:\:C=80\\\\Bentuk\:\:\:persamaan\:\:\:kuadrat\\Ax^2+Bx=-C\\x^2-21x=-80\\x^2-2\left(\frac{21}{2}\right)x+\left(\frac{21}{2}\right)^2=-80+\left(\frac{21}{2}\right)^2\\\because u^2-2uv+v^2=(u-v)^2\therefore[/tex]
[tex]\displaystyle\left(x-\frac{21}{2}\right)^2=-80+\left(\frac{21}{2}\right)^2\\\\\left(x-\frac{21}{2}\right)^2=-\frac{320}{4}+\frac{441}{4}\\\\\left(x-\frac{21}{2}\right)^2=\frac{121}{4}\\\\\left|x-\frac{21}{2}\right|=\sqrt{\frac{121}{4}}\\\\\left|x-\frac{21}{2}\right|=\frac{\sqrt{121}}{\sqrt{4}}\\\\\left|x-\frac{21}{2}\right|=\frac{11}{2}\\\\x=\left\{\frac{11}{2}+\frac{21}{2},\:\:\:\:-\frac{11}{2}+\frac{21}{2}\right\}\\\\p=\frac{11}{2}+\frac{21}{2},\:\:\:q=-\frac{11}{2}+\frac{21}{2}[/tex]
[tex]\displaystyle p=\frac{32}{2},\:\:\:q=\frac{10}{2}[/tex]
p = 16, q = 5
Oleh karena itu hasil sederhananya
[tex]\sqrt{p}+\sqrt{q}=\sqrt{16}+\sqrt{5}=\\\bf4+\sqrt{5}[/tex]
(ii) Hitunglah
[tex]\sqrt{75} -2\sqrt{48} -\sqrt{147} +3\sqrt{27} \\=\sqrt{25(3)} -2\sqrt{16(3)} -\sqrt{49(3)} +3\sqrt{9(3)} \\=\sqrt{25}\sqrt{3} -2\sqrt{16}\sqrt{3} -\sqrt{49}\sqrt{3} +3\sqrt{9}\sqrt{3}\\=5\sqrt{3} -2\cdot4\sqrt{3} -7\sqrt{3} +3\cdot3\sqrt{3}\\=5\sqrt{3} -8\sqrt{3} -7\sqrt{3} +9\sqrt{3}\\=(5-8-7+9)\sqrt[3}\\=-\sqrt{3}[/tex]
(iii) Bentuk baru dari (8,6×10⁸) × (4×10⁶)
= (8,6 × 4) × (10⁸ × 10⁶)
= 34,4 × (10⁸ × 10⁶)
= 34,4 ÷ 10 × 10 × (10⁸ × 10⁶)
= 3,44 × (10¹ × 10⁸ × 10⁶)
∵ nᵃ × nᵇ = nᵃ⁺ᵇ ∴
= 3,44 × 10¹⁺⁸⁺⁶
= 3,44 × 10¹⁵
(xcvi)