Soal Bentuk Akar dan Pembahasan

Hallo...! Pengunjung setia Catatan Matematika, kali ini Bang RP (Reikson Panjaitan, S.Pd) berbagi Kumpulan Soal Bentuk Akar berserta pembahasannya. Ayo... manfaatkan website Catatan Matematika ini untuk belajar matematika secara online.

Tata Cara Belajar:
Cobalah mengerjakan soal-soal yang tersedia secara mandiri. Setelah itu cek jawaban kamu dengan pembahasan yang telah disediakan, dengan cara:
klik "LIHAT/TUTUP:".

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Soal No. 1

Hasil dari $\sqrt{75}-\sqrt{12}$ = ….
A. $\sqrt{3}$
B. $2\sqrt{3}$
C. $3\sqrt{3}$
D. $4\sqrt{3}$
E. $5\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\sqrt{75}-\sqrt{12}$
= $\sqrt{25\times 3}-\sqrt{4\times 3}$
= $5\sqrt{3}-2\sqrt{3}$
= $3\sqrt{3}$
Jawaban: C

Soal No. 2

Bentuk sederhana dari $2\sqrt{18}-\sqrt{8}+\sqrt{2}$ adalah ....
A. $3\sqrt{2}$
B. $4\sqrt{3}-\sqrt{2}$
C. $5\sqrt{2}$
D. $4\sqrt{3}+\sqrt{2}$
E. $17\sqrt{2}$

Penyelesaian: Lihat/Tutup

$2\sqrt{18}-\sqrt{8}+\sqrt{2}$
= $2\sqrt{9\times 2}-\sqrt{4\times 2}+\sqrt{2}$
= $2.3\sqrt{2}-2\sqrt{2}+\sqrt{2}$
= $6\sqrt{2}-2\sqrt{2}+\sqrt{2}$
= $5\sqrt{2}$
Jawaban: C

Soal No. 3

Hasil dari $3\sqrt{8}-\sqrt{50}+2\sqrt{18}$ = ….
A. $7\sqrt{2}$
B. $13\sqrt{2}$
C. $14\sqrt{2}$
D. $20\sqrt{2}$
E. $23\sqrt{2}$

Penyelesaian: Lihat/Tutup

$3\sqrt{8}-\sqrt{50}+2\sqrt{18}$
= $3\sqrt{4\times 2}-\sqrt{25\times 2}+2\sqrt{9\times 2}$
= $3.2\sqrt{2}-5\sqrt{2}+2.3\sqrt{2}$
= $6\sqrt{2}-5\sqrt{2}+6\sqrt{2}$
= $7\sqrt{2}$
Jawaban: A

Soal No. 4

Hasil dari $3\sqrt{27}-2\sqrt{48}+6\sqrt{75}$ = ….
A. $12\sqrt{3}$
B. $14\sqrt{3}$
C. $28\sqrt{3}$
D. $30\sqrt{3}$
E. $31\sqrt{3}$

Penyelesaian: Lihat/Tutup

$3\sqrt{27}-2\sqrt{48}+6\sqrt{75}$
= $3\sqrt{9\times 3}-2\sqrt{16\times 3}+6\sqrt{25\times 3}$
= $3.3\sqrt{3}-2.4\sqrt{3}+6.5\sqrt{3}$
= $9\sqrt{3}-8\sqrt{3}+30\sqrt{3}$
= $31\sqrt{3}$
Jawaban: E

Soal No. 5

Hasil dari $\sqrt{50}-\sqrt{108}+2\sqrt{12}+\sqrt{32}$ adalah ….
A. $7\sqrt{2}-2\sqrt{3}$
B. $13\sqrt{2}-14\sqrt{3}$
C. $9\sqrt{2}-4\sqrt{3}$
D. $9\sqrt{2}-2\sqrt{3}$
E. $13\sqrt{2}-2\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\sqrt{50}-\sqrt{108}+2\sqrt{12}+\sqrt{32}$
= $\sqrt{25\times 2}-\sqrt{36\times 3}+2\sqrt{4\times 3}+\sqrt{16\times 2}$
= $5\sqrt{2}-6\sqrt{3}+2.2\sqrt{3}+4\sqrt{2}$
= $5\sqrt{2}-6\sqrt{3}+4\sqrt{3}+4\sqrt{2}$
= $9\sqrt{2}-2\sqrt{3}$
Jawaban: D

Soal No. 6

Hasil dari $\sqrt{2}-\sqrt{8}+\sqrt{27}+\sqrt{50}-\sqrt{75}$ = ….
A. $3\sqrt{3}$
B. $3\sqrt{3}-2$
C. $2\sqrt{3}$
D. $\sqrt{3}-\sqrt{6}$
E. $4\sqrt{2}-2\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\sqrt{2}-\sqrt{8}+\sqrt{27}+\sqrt{50}-\sqrt{75}$
= $\sqrt{2}-\sqrt{4\times 2}$+$\sqrt{9\times 3}$+$\sqrt{25\times 2}-\sqrt{25\times 3}$
= $\sqrt{2}-2\sqrt{2}+3\sqrt{3}+5\sqrt{2}-5\sqrt{3}$
= $4\sqrt{2}-2\sqrt{3}$
Jawaban: E

Soal No. 7

Hasil dari $\sqrt{2}\times \sqrt{3}\times \sqrt{48}:6\sqrt{2}$ = ….
A. $3\sqrt{2}$
B. $2\sqrt{2}$
C. 3
D. 2
E. 1

Penyelesaian: Lihat/Tutup

$\sqrt{2}\times \sqrt{3}\times \sqrt{48}:6\sqrt{2}$
= $\frac{\sqrt{2}\times \sqrt{3}\times \sqrt{48}}{6\sqrt{2}}$
= $\frac{\sqrt{3}\times \sqrt{48}}{6}$
= $\frac{\sqrt{3}\times \sqrt{16\times 3}}{6}$
= $\frac{\sqrt{3}\times 4\sqrt{3}}{6}$
= $\frac{4\times 3}{6}$
= 2
Jawaban: D

Soal No. 8

Hasil dari $(2+3\sqrt{3})-(5-2\sqrt{75})$ adalah ….
A. $-7\sqrt{3}-3$
B. $-7\sqrt{3}+3$
C. $13\sqrt{3}-7$
D. $13\sqrt{3}-3$
E. $13\sqrt{3}+3$

Penyelesaian: Lihat/Tutup

$(2+3\sqrt{3})-(5-2\sqrt{75})$
= $2+3\sqrt{3}-5+2\sqrt{75}$
= $-3+3\sqrt{3}+2\sqrt{25\times 3}$
= $-3+3\sqrt{3}+2.5\sqrt{3}$
= $-3+3\sqrt{3}+10\sqrt{3}$
= $-3+13\sqrt{3}$
= $13\sqrt{3}-3$
Jawaban: D

Soal No. 9

Hasil dari $(2\sqrt{2}-\sqrt{6})(\sqrt{2}+\sqrt{6})$ = ….
A. $2(1-\sqrt{2})$
B. $2(2-\sqrt{2})$
C. $2(\sqrt{3}-1)$
D. $3(\sqrt{3}-1)$
E. $4(2\sqrt{3}+1)$

Penyelesaian: Lihat/Tutup

$(2\sqrt{2}-\sqrt{6})(\sqrt{2}+\sqrt{6})$
= $2\sqrt{2}.\sqrt{2}+2\sqrt{2}.\sqrt{6}-\sqrt{6}.\sqrt{2}-\sqrt{6}.\sqrt{6}$
= $2.2+2\sqrt{12}-\sqrt{12}-6$
= $4+\sqrt{12}-6$
= $\sqrt{4\times 3}-2$
= $2\sqrt{3}-2$
= $2(\sqrt{3}-1)$
Jawaban: C

Soal No. 10

Hasil dari $(5\sqrt{3}+7\sqrt{2})(6\sqrt{3}-4\sqrt{2})$ = ….
A. $22-24\sqrt{3}$
B. $34-22\sqrt{3}$
C. $22+34\sqrt{6}$
D. $34+22\sqrt{6}$
E. $146+22\sqrt{6}$

Penyelesaian: Lihat/Tutup

$(5\sqrt{3}+7\sqrt{2})(6\sqrt{3}-4\sqrt{2})$
= $5\sqrt{3}.6\sqrt{3}-5\sqrt{3}.4\sqrt{2}$+$7\sqrt{2}.6\sqrt{3}-7\sqrt{2}.4\sqrt{2}$
= $90-20\sqrt{6}+42\sqrt{6}-56$
= $34+22\sqrt{6}$
Jawaban: D

Soal No. 11

Hasil dari $(3\sqrt{6}+4\sqrt{2})(5\sqrt{6}-3\sqrt{2})$ = ….
A. $66-46\sqrt{3}$
B. $66-22\sqrt{3}$
C. $66+22\sqrt{3}$
D. $66+46\sqrt{3}$
E. $114+22\sqrt{3}$

Penyelesaian: Lihat/Tutup

$(3\sqrt{6}+4\sqrt{2})(5\sqrt{6}-3\sqrt{2})$
= $3\sqrt{6}.5\sqrt{6}-3\sqrt{6}.3\sqrt{2}$+$4\sqrt{2}.5\sqrt{6}-4\sqrt{2}.3\sqrt{2}$
= $90-9\sqrt{12}+20\sqrt{12}-24$
= $66+11\sqrt{12}$
= $66+11\sqrt{4\times 3}$
= $66+11.2\sqrt{3}$
= $66+22\sqrt{3}$
Jawaban: C

Soal No. 12

Hasil dari $\frac{5}{2\sqrt{3}}$ adalah ….
A. $\frac{5}{3}\sqrt{3}$
B. $\sqrt{3}$
C. $\frac{5}{6}\sqrt{3}$
D. $\frac{5}{9}\sqrt{3}$
E. $\frac{5}{12}\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{5}{2\sqrt{3}} &= \frac{5}{2\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}} \\ &= \frac{5\sqrt{3}}{2.3} \\ &= \frac{5}{6}\sqrt{3} \end{align}$
Jawaban: C

Soal No. 13

Bentuk sederhana dari $\frac{4}{3\sqrt{5}}$ adalah ….
A. $\frac{1}{5}\sqrt{5}$
B. $\frac{1}{15}\sqrt{5}$
C. $\frac{2}{15}\sqrt{5}$
D. $\frac{4}{15}\sqrt{5}$
E. $\frac{4}{15}\sqrt{15}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{4}{3\sqrt{5}} &= \frac{4}{3\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}} \\ &= \frac{4\sqrt{5}}{3.5} \\ &= \frac{4}{15}\sqrt{5} \end{align}$
Jawaban: E

Soal No. 14

Bentuk sederhana $\frac{2}{3-\sqrt{7}}$ adalah ….
A. $6+2\sqrt{7}$
B. $6-2\sqrt{7}$
C. $3+\sqrt{7}$
D. $3-\sqrt{7}$
E. $-3-\sqrt{7}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{2}{3-\sqrt{7}} &= \frac{2}{3-\sqrt{7}}\times \frac{3+\sqrt{7}}{3+\sqrt{7}} \\ &= \frac{2(3+\sqrt{7})}{9-7} \\ &= \frac{2(3+\sqrt{7})}{2} \\ &= 3+\sqrt{7} \end{align}$
Jawaban: C

Soal No. 15

Bentuk sederhana dari $\frac{7}{3+\sqrt{2}}$ adalah ….
A. $21+7\sqrt{2}$
B. $21+\sqrt{2}$
C. $21-7\sqrt{2}$
D. $3+\sqrt{2}$
E. $3-\sqrt{2}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{7}{3+\sqrt{2}} &= \frac{7}{3+\sqrt{2}}\times \frac{3-\sqrt{2}}{3-\sqrt{2}} \\ &= \frac{7(3-\sqrt{2})}{9-2} \\ &= \frac{7(3-\sqrt{2})}{7} \\ &= 3-\sqrt{2} \end{align}$
Jawaban: E

Soal No. 16

Bentuk sederhana dari $\frac{4}{3+\sqrt{5}}$ adalah ….
A. $3+\sqrt{5}$
B. $3-\sqrt{5}$
C. $\sqrt{5}-3$
D. $\sqrt{5}+4$
E. $4+\sqrt{5}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{4}{3+\sqrt{5}} &= \frac{4}{3+\sqrt{5}}\times \frac{3-\sqrt{5}}{3-\sqrt{5}} \\ &= \frac{4(3-\sqrt{5})}{9-5} \\ &= \frac{4(3-\sqrt{5})}{4} \\ &= 3-\sqrt{5} \end{align}$
Jawaban: B

Soal No. 17

Bentuk sederhana dari $\frac{6}{4+\sqrt{5}}$ adalah ….
A. $\frac{2}{3}(4+\sqrt{5})$
B. $\frac{6}{11}(4+\sqrt{5})$
C. $\frac{6}{11}(4-\sqrt{5})$
D. $\frac{6}{11}(-4+\sqrt{5})$
E. $\frac{2}{3}(-4+\sqrt{5})$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{6}{4+\sqrt{5}} &= \frac{6}{4+\sqrt{5}}\times \frac{4-\sqrt{5}}{4-\sqrt{5}} \\ &= \frac{6(4-\sqrt{5})}{16-5} \\ &= \frac{6}{11}(4-\sqrt{5}) \end{align}$
Jawaban: C

Soal No. 18

Bentuk sederhana dari $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ adalah ….
A. $4-2\sqrt{15}$
B. $4-\sqrt{15}$
C. $4+\sqrt{15}$
D. $4+2\sqrt{15}$
E. $8+2\sqrt{15}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} &= \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}} \\ &= \frac{5+\sqrt{15}+\sqrt{15}+3}{5-3} \\ &= \frac{8+2\sqrt{15}}{2} \\ &= 4+\sqrt{15} \end{align}$
Jawaban: C

Soal No. 19

Dengan merasionalkan penyebut, bentuk sederhana dari $\frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}-\sqrt{5}}$ adalah ….
A. $11+\sqrt{30}$
B. $11+2\sqrt{30}$
C. $1+\sqrt{30}$
D. $1+2\sqrt{30}$
E. $2\sqrt{30}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}-\sqrt{5}} &= \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}-\sqrt{5}}\times \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}} \\ &= \frac{6+\sqrt{30}+\sqrt{30}+5}{6-5} \\ &= 11+2\sqrt{30} \end{align}$
Jawaban: B

Soal No. 20

Bentuk sederhana dari $\frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}-\sqrt{2}}$ adalah ….
A. $1+\frac{1}{2}\sqrt{3}$
B. $\frac{1}{2}+\sqrt{3}$
C. $2+\frac{1}{2}\sqrt{3}$
D. $2+\sqrt{3}$
E. $1+2\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}-\sqrt{2}} &= \frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}-\sqrt{2}}\times \frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}} \\ &= \frac{6+\sqrt{12}+\sqrt{12}+2}{6-2} \\ &= \frac{8+2\sqrt{12}}{4} \\ &= \frac{8+2\sqrt{4\times 3}}{4} \\ &= \frac{8+2.2\sqrt{3}}{4} \\ &= \frac{8+4\sqrt{3}}{4} \\ &= 2+\sqrt{3} \end{align}$
Jawaban: D

Soal No. 21

Bentuk sederhana dari $\frac{\sqrt{15}+\sqrt{5}}{\sqrt{15}-\sqrt{5}}$ adalah ….
A. $20+\sqrt{3}$
B. $2+10\sqrt{3}$
C. $1+10\sqrt{3}$
D. $2+\sqrt{3}$
E. $1+\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{15}+\sqrt{5}}{\sqrt{15}-\sqrt{5}} &= \frac{\sqrt{15}+\sqrt{5}}{\sqrt{15}-\sqrt{5}}\times \frac{\sqrt{15}+\sqrt{5}}{\sqrt{15}+\sqrt{5}} \\ &= \frac{15+\sqrt{75}+\sqrt{75}+5}{15-5} \\ &= \frac{20+2\sqrt{75}}{10} \\ &= \frac{20+2\sqrt{25\times 3}}{10} \\ &= \frac{20+2.5\sqrt{3}}{10} \\ &= \frac{20+10\sqrt{3}}{10} \\ &= 2+\sqrt{3} \end{align}$
Jawaban: D

Soal No. 22

Bentuk sederhana dari $\frac{\sqrt{27}-\sqrt{45}}{\sqrt{3}-\sqrt{5}}$ adalah ….
A. 1
B. $\sqrt{7}$
C. 3
D. $\sqrt{14}$
E. 5

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{27}-\sqrt{45}}{\sqrt{3}-\sqrt{5}} &= \frac{\sqrt{9\times 3}-\sqrt{9\times 5}}{\sqrt{3}-\sqrt{5}} \\ &= \frac{3\sqrt{3}-3\sqrt{5}}{\sqrt{3}-\sqrt{5}} \\ &= \frac{3(\sqrt{3}-\sqrt{5})}{(\sqrt{3}-\sqrt{5})} \\ &= 3 \end{align}$
Jawaban: C

Soal No. 23

Hasil dari $\sqrt{12}+\sqrt{27}-\sqrt{3}$ adalah ….
A. 6
B. $4\sqrt{3}$
C. $5\sqrt{3}$
D. $6\sqrt{3}$
E. $12\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\sqrt{12}+\sqrt{27}-\sqrt{3}$
= $\sqrt{4\times 3}+\sqrt{9\times 3}-\sqrt{3}$
= $2\sqrt{3}+3\sqrt{3}-\sqrt{3}$
= $4\sqrt{3}$
Jawaban: B

Soal No. 24

Bentuk sederhana dari $\sqrt{8}+\sqrt{75}-(\sqrt{32}+\sqrt{243})$ adalah ….
A. $2\sqrt{2}+14\sqrt{3}$
B. $-2\sqrt{2}-4\sqrt{3}$
C. $-2\sqrt{2}+4\sqrt{3}$
D. $-2\sqrt{2}+14\sqrt{3}$
E. $2\sqrt{2}-4\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\sqrt{8}+\sqrt{75}-(\sqrt{32}+\sqrt{243})$
= $\sqrt{8}+\sqrt{75}-\sqrt{32}-\sqrt{243}$
= $\sqrt{4\times 2}+\sqrt{25\times 3}-\sqrt{16\times 2}-\sqrt{81\times 3}$
= $2\sqrt{2}+5\sqrt{3}-4\sqrt{2}-9\sqrt{3}$
= $-2\sqrt{2}-4\sqrt{3}$
Jawaban: B

Soal No. 25

Bentuk sederhana dari $\frac{24}{3-\sqrt{7}}$ adalah ….
A. $18-24\sqrt{7}$
B. $18-6\sqrt{7}$
C. $12+4\sqrt{7}$
D. $18+6\sqrt{7}$
E. $36+12\sqrt{7}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{24}{3-\sqrt{7}} &= \frac{24}{3-\sqrt{7}}\times \frac{3+\sqrt{7}}{3+\sqrt{7}} \\ &= \frac{24(3+\sqrt{7})}{9-7} \\ &= \frac{24(3+\sqrt{7})}{2} \\ &= 12(3+\sqrt{7}) \\ &= 36+12\sqrt{7} \end{align}$
Jawaban: E

Soal No. 26

Bentuk sederhana dari $\frac{\sqrt{5}+2\sqrt{3}}{\sqrt{5}-3\sqrt{3}}$ = ….
A. $\frac{20+5\sqrt{15}}{22}$
B. $\frac{23-5\sqrt{15}}{22}$
C. $\frac{20-5\sqrt{15}}{-22}$
D. $\frac{20+5\sqrt{15}}{-22}$
E. $\frac{23+5\sqrt{15}}{-22}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{5}+2\sqrt{3}}{\sqrt{5}-3\sqrt{3}} &= \frac{\sqrt{5}+2\sqrt{3}}{\sqrt{5}-3\sqrt{3}}\times \frac{\sqrt{5}+3\sqrt{3}}{\sqrt{5}+3\sqrt{3}} \\ &= \frac{5+3\sqrt{15}+2\sqrt{15}+18}{5-27} \\ &= \frac{23+5\sqrt{15}}{-22} \end{align}$
Jawaban: E

Soal No. 27

Bentuk sederhana dari $\frac{\sqrt{3}+3\sqrt{2}}{\sqrt{3}-6\sqrt{2}}$ = ….
A. $-\frac{1}{23}(13+3\sqrt{6})$
B. $-\frac{1}{23}(13-3\sqrt{6})$
C. $-\frac{1}{23}(-11-\sqrt{6})$
D. $\frac{1}{23}(11+3\sqrt{6})$
E. $\frac{1}{23}(13+3\sqrt{6})$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{3}+3\sqrt{2}}{\sqrt{3}-6\sqrt{2}} &= \frac{\sqrt{3}+3\sqrt{2}}{\sqrt{3}-6\sqrt{2}}\times \frac{\sqrt{3}+6\sqrt{2}}{\sqrt{3}+6\sqrt{2}} \\ &= \frac{3+6\sqrt{6}+3\sqrt{6}+36}{3-72} \\ &= \frac{39+9\sqrt{6}}{-69} \\ &= \frac{13+3\sqrt{6}}{-23} \\ &= -\frac{1}{23}(13+3\sqrt{6}) \end{align}$
Jawaban: A

Soal No. 28

Bentuk sederhana dari $\frac{4(2+\sqrt{3})(2-\sqrt{3})}{3+\sqrt{5}}$ = ….
A. $-(3-\sqrt{5})$
B. $-\frac{1}{4}(3-\sqrt{5})$
C. $\frac{1}{4}(3-\sqrt{5})$
D. $3-\sqrt{5}$
E. $3+\sqrt{5}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{4(2+\sqrt{3})(2-\sqrt{3})}{3+\sqrt{5}} &= \frac{4(4-3)}{3+\sqrt{5}} \\ &= \frac{4}{3+\sqrt{5}}\times \frac{3-\sqrt{5}}{3-\sqrt{5}} \\ &= \frac{4(3-\sqrt{5})}{9-5} \\ &= \frac{4(3-\sqrt{5})}{4} \\ &= 3-\sqrt{5} \end{align}$
Jawaban: D

Soal No. 29

Bentuk sederhana dari $\frac{6(3+\sqrt{5})(3-\sqrt{5})}{2+\sqrt{6}}$ = ….
A. $24+12\sqrt{6}$
B. $-24+12\sqrt{6}$
C. $24-12\sqrt{6}$
D. $-24-\sqrt{6}$
E. $-24-12\sqrt{6}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{6(3+\sqrt{5})(3-\sqrt{5})}{2+\sqrt{6}} &= \frac{6(9-5)}{2+\sqrt{6}} \\ &= \frac{24}{2+\sqrt{6}}\times \frac{2-\sqrt{6}}{2-\sqrt{6}} \\ &= \frac{24(2-\sqrt{6})}{4-6} \\ &= \frac{48-24\sqrt{6}}{-2} \\ &= -24+12\sqrt{6} \end{align}$
Jawaban: B

Soal No. 30

Bentuk sederhana dari $\frac{\sqrt{2}+3\sqrt{5}}{\sqrt{2}-\sqrt{5}}$ adalah ….
A. $\frac{1}{3}(17-4\sqrt{10})$
B. $-\frac{2}{3}(15-4\sqrt{10})$
C. $\frac{2}{3}(15-4\sqrt{10})$
D. $-\frac{1}{3}(17-4\sqrt{10})$
E. $-\frac{1}{3}(17+4\sqrt{10})$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{2}+3\sqrt{5}}{\sqrt{2}-\sqrt{5}} &= \frac{\sqrt{2}+3\sqrt{5}}{\sqrt{2}-\sqrt{5}}\times \frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}+\sqrt{5}} \\ &= \frac{2+\sqrt{10}+3\sqrt{10}+15}{2-5} \\ &= \frac{17+4\sqrt{10}}{-3} \\ &= -\frac{1}{3}(17+4\sqrt{10}) \end{align}$
Jawaban: E

Soal No. 31

Bentuk $\frac{3\sqrt{3}+\sqrt{7}}{\sqrt{7}-2\sqrt{3}}$ dapat disederhanakan menjadi bentuk ….
A. $-5-\sqrt{21}$
B. $-5+\sqrt{21}$
C. $-5+5\sqrt{21}$
D. $-25+5\sqrt{21}$
E. $-25-5\sqrt{21}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{3\sqrt{3}+\sqrt{7}}{\sqrt{7}-2\sqrt{3}} &= \frac{3\sqrt{3}+\sqrt{7}}{\sqrt{7}-2\sqrt{3}}\times \frac{\sqrt{7}+2\sqrt{3}}{\sqrt{7}+2\sqrt{3}} \\ &= \frac{3\sqrt{21}+18+7+2\sqrt{21}}{7-12} \\ &= \frac{25+5\sqrt{21}}{-5} \\ &= -5-\sqrt{21} \end{align}$
Jawaban: A

Soal No. 32

Bentuk sederhana dari $\frac{\sqrt{2}-2\sqrt{3}}{\sqrt{2}-\sqrt{3}}$ = ….
A. $4+\sqrt{6}$
B. $4-\sqrt{6}$
C. $-4+\sqrt{6}$
D. $-4-\sqrt{6}$
E. $-4-3\sqrt{6}$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{2}-2\sqrt{3}}{\sqrt{2}-\sqrt{3}} &= \frac{\sqrt{2}-2\sqrt{3}}{\sqrt{2}-\sqrt{3}}\times \frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}+\sqrt{3}} \\ &= \frac{2+\sqrt{6}-2\sqrt{6}-6}{2-3} \\ &= \frac{-4-\sqrt{6}}{-1} \\ &= 4+\sqrt{6} \end{align}$
Jawaban: A

Soal No. 33

Bentuk sederhana dari $\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+3\sqrt{2}}$ adalah ….
A. $-\frac{1}{13}(-11+4\sqrt{10})$
B. $-\frac{1}{13}(-1+4\sqrt{10})$
C. $\frac{1}{13}(11-4\sqrt{10})$
D. $-\frac{1}{13}(11+4\sqrt{10})$
E. $\frac{1}{13}(-11+4\sqrt{10})$

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+3\sqrt{2}} &= \frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+3\sqrt{2}}\times \frac{\sqrt{5}-3\sqrt{2}}{\sqrt{5}-3\sqrt{2}} \\ &= \frac{5-3\sqrt{10}-\sqrt{10}+6}{5-18} \\ &= \frac{11-4\sqrt{10}}{-13} \\ &= \frac{-11+4\sqrt{10}}{13} \\ &= \frac{1}{13}(-11+4\sqrt{10}) \end{align}$
Jawaban: E

Soal No. 34

$\sqrt{12+\sqrt{12+\sqrt{12+...}}}=x$, nilai $x$ adalah ….
A. 3
B. 4
C. 6
D. 8
E. 12

Penyelesaian: Lihat/Tutup

$\begin{align}x &= \sqrt{12+\sqrt{12+\sqrt{12+...}}} \\ x &= \sqrt{12+x} \\ x^2 &= 12+x \\ x^2-x-12 &= 0 \\ (x+3)(x-4) &= 0 \end{align}$
$x=-3$ atau $x=4$
Karena $x=\sqrt{12+\sqrt{12+\sqrt{12+...}}}$ maka $x>0$, jadi nilai $x$ yang memenuhi adalah 4.
Jawaban: B

Soal No. 35

Bentuk $\sqrt{52-14\sqrt{3}}$ senilai dengan ….
A. $7-\sqrt{3}$
B. $7+\sqrt{3}$
C. $\sqrt{7}+3$
D. $\sqrt{7}-3$
E. $\sqrt{7}-\sqrt{3}$

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)-2\sqrt{ab}}=\sqrt{a}-\sqrt{b}$ dengan $a > b$ maka:
$\begin{align}\sqrt{52-14\sqrt{3}} &= \sqrt{52-2.7\sqrt{3}} \\ &= \sqrt{52-2\sqrt{49\times 3}} \\ &= \sqrt{(49+3)-2\sqrt{49\times 3}} \\ &= \sqrt{49}-\sqrt{3} \\ &= 7-\sqrt{3} \end{align}$
Jawaban: A

Soal No. 36

$\sqrt{5+2\sqrt{6}}$ = ….
A. $1+\sqrt{6}$
B. $1-\sqrt{6}$
C. $\sqrt{3}+\sqrt{2}$
D. $\sqrt{3}-\sqrt{2}$
E. $\sqrt{5}+1$

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}$ dengan $a > b$ maka:
$\begin{align}\sqrt{5+2\sqrt{6}} &= \sqrt{(3+2)+2\sqrt{3\times 2}} \\ &= \sqrt{3}+\sqrt{2} \end{align}$
Jawaban: C

Soal No. 37

$\sqrt{8+2\sqrt{15}}$ = ….
A. $3+\sqrt{5}$
B. $3-\sqrt{5}$
C. $\sqrt{5}+\sqrt{3}$
D. $\sqrt{5}-\sqrt{3}$
E. $1+\sqrt{15}$

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}$ dengan $a > b$ maka:
$\begin{align}\sqrt{8+2\sqrt{15}} &= \sqrt{(5+3)+2\sqrt{5\times 3}} \\ &= \sqrt{5}+\sqrt{3} \end{align}$
Jawaban: C

Soal No. 38

$\sqrt{19+8\sqrt{3}}$ = ….
A. $5+\sqrt{3}$
B. $5-\sqrt{3}$
C. $4+\sqrt{3}$
D. $4-\sqrt{3}$
E. $2+\sqrt{3}$

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}$ dengan $a > b$ maka:
$\begin{align}\sqrt{19+8\sqrt{3}} &= \sqrt{19+2.4\sqrt{3}} \\ &= \sqrt{19+2\sqrt{16\times 3}} \\ &= \sqrt{(16+3)+2\sqrt{16\times 3}} \\ &= \sqrt{16}+\sqrt{3} \\ &= 4+\sqrt{3} \end{align}$
Jawaban: C

Soal No. 39

$\sqrt{14-8\sqrt{3}}$ = ….
A. $4+\sqrt{6}$
B. $4-\sqrt{6}$
C. $\sqrt{8}-\sqrt{6}$
D. $\sqrt{6}-\sqrt{8}$
E. C dan D benar

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)-2\sqrt{ab}}=\sqrt{a}-\sqrt{b}$ dengan $a > b$ maka:
$\begin{align}\sqrt{14-8\sqrt{3}} &= \sqrt{14-2.4\sqrt{3}} \\ &= \sqrt{14-2\sqrt{16\times 3}} \\ &= \sqrt{14-2\sqrt{48}} \\ &= \sqrt{(8+6)-2\sqrt{8\times 6}} \\ &= \sqrt{8}-\sqrt{6} \end{align}$
Jawaban: C

Soal No. 40

$\sqrt{27+10\sqrt{2}}$ = ….
A. $5+\sqrt{2}$
B. $5-\sqrt{2}$
C. $4+\sqrt{2}$
D. $4-\sqrt{2}$
E. $3+\sqrt{2}$

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}$ dengan $a > b$ maka:
$\begin{align}\sqrt{27+10\sqrt{2}} &= \sqrt{27+2.5\sqrt{2}} \\ &= \sqrt{27+2\sqrt{25\times 2}} \\ &= \sqrt{(25+2)+2\sqrt{25\times 2}} \\ &= \sqrt{25}+\sqrt{2} \\ &= 5+\sqrt{2} \end{align}$
Jawaban: A

Soal No. 41

$\sqrt{5+\sqrt{24}}$ = ….
A. $\sqrt{6}-\sqrt{2}$
B. $\sqrt{3}-\sqrt{2}$
C. $\sqrt{6}-\sqrt{3}$
D. $\sqrt{6}+\sqrt{2}$
E. $\sqrt{3}+\sqrt{2}$

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}$ dengan $a > b$ maka:
$\begin{align}\sqrt{5+\sqrt{24}} &= \sqrt{5+\sqrt{4\times 6}} \\ &= \sqrt{5+2\sqrt{6}} \\ &= \sqrt{(3+2)+2\sqrt{3\times 2}} \\ &= \sqrt{3}+\sqrt{2} \end{align}$
Jawaban: E

Soal No. 42

$\sqrt{3-\sqrt{5}}$ = ….
A. $\frac{1}{2}(3-\sqrt{5})$
B. $\frac{1}{2}(\sqrt{20}-\sqrt{2})$
C. $\frac{1}{2}(\sqrt{20}-\sqrt{3})$
D. $\frac{1}{2}(\sqrt{10}-\sqrt{2})$
E. $\frac{1}{2}(\sqrt{5}-\sqrt{2})$

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)-2\sqrt{ab}}=\sqrt{a}-\sqrt{b}$ dengan $a > b$ maka:
$\begin{align}\sqrt{3-\sqrt{5}} &= \sqrt{\frac{1}{2}(6-2\sqrt{5})} \\ &= \sqrt{\frac{1}{2}}\times \sqrt{6-2\sqrt{5}} \\ &= \frac{1}{\sqrt{2}}\times \sqrt{(5+1)-2\sqrt{5\times 1}} \\ &= \frac{1}{\sqrt{2}}(\sqrt{5}-\sqrt{1}) \\ &= \frac{\sqrt{5}-1}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}} \\ &= \frac{\sqrt{10}-\sqrt{2}}{2} \\ &= \frac{1}{2}(\sqrt{10}-\sqrt{2}) \end{align}$
Jawaban: D

Soal No. 43

$\sqrt{14-3\sqrt{20}}=a-\sqrt{b}$ maka $a.b$ = ….
A. $-15$
B. $-10$
C. $-5$
D. 10
E. 15

Penyelesaian: Lihat/Tutup

$\begin{align}\sqrt{14-3\sqrt{20}} &= a-\sqrt{b} \\ \sqrt{14-3\sqrt{4.5}} &= a-\sqrt{b} \\ \sqrt{14-3.2\sqrt{5}} &= a-\sqrt{b} \\ \sqrt{14-2\sqrt{9\times 5}} &= a-\sqrt{b} \\ \sqrt{(9+5)-2\sqrt{9\times 5}} &= a-\sqrt{b} \\ \sqrt{9}-\sqrt{5} &= a-\sqrt{b} \\ 3-\sqrt{5} &= a-\sqrt{b} \end{align}$
$a=3$ dan $b=5$ maka $a.b=3.5=15$
Jawaban: E

Soal No. 44

$\sqrt{6-3\sqrt{3}}=a\sqrt{2}+b\sqrt{6}$ maka $1-\frac{a}{b}$ = ….
A. $-3$
B. $-1$
C. 1
D. 3
E. 5

Penyelesaian: Lihat/Tutup

$\begin{align}\sqrt{6-3\sqrt{3}} &= a\sqrt{2}+b\sqrt{6} \\ \sqrt{\frac{1}{2}(12-6\sqrt{3})} &= a\sqrt{2}+b\sqrt{6} \\ \sqrt{\frac{1}{2}(12-2.3\sqrt{3})} &= a\sqrt{2}+b\sqrt{6} \\ \sqrt{\frac{1}{2}}\times \sqrt{12-2\sqrt{9\times 3}} &= a\sqrt{2}+b\sqrt{6} \\ \frac{1}{\sqrt{2}}\times \sqrt{(9+3)-2\sqrt{9\times 3}} &= a\sqrt{2}+b\sqrt{6} \\ \frac{1}{\sqrt{2}}(\sqrt{9}-\sqrt{3}) &= a\sqrt{2}+b\sqrt{6} \\ \frac{2-\sqrt{3}}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}} &= a\sqrt{2}+b\sqrt{6} \\ \frac{2\sqrt{2}-\sqrt{6}}{2} &= a\sqrt{2}+b\sqrt{6} \\ \sqrt{2}-\frac{1}{2}\sqrt{6} &= a\sqrt{2}+b\sqrt{6} \end{align}$
$a=1$ dan $b=-\frac{1}{2}$ maka:
$1-\frac{a}{b}=1-\frac{1}{-\frac{1}{2}}=1-(-2)=3$
Jawaban: D

Soal No. 45

${{\left( \sqrt{13+4\sqrt{3}}+3 \right)}^{\frac{1}{2}}}$ = ….
A. $1+\sqrt{3}$
B. $1-\sqrt{3}$
C. $1-2\sqrt{3}$
D. $1+2\sqrt{3}$
E. $2+\sqrt{3}$

Penyelesaian: Lihat/Tutup

Ingat, $\sqrt{(a+b)+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}$ dengan $a > b$ maka:
${{\left( \sqrt{13+4\sqrt{3}}+3 \right)}^{\frac{1}{2}}}$
= $\sqrt{\sqrt{13+2.2\sqrt{3}}+3}$
= $\sqrt{\sqrt{13+2\sqrt{4\times 3}}+3}$
= $\sqrt{\sqrt{13+2\sqrt{12}}+3}$
= $\sqrt{\sqrt{(12+1)+2\sqrt{12\times 1}}+3}$
= $\sqrt{\sqrt{12}+\sqrt{1}+3}$
= $\sqrt{4+\sqrt{12}}$
= $\sqrt{4+\sqrt{4\times 3}}$
= $\sqrt{4+2\sqrt{3}}$
= $\sqrt{(3+1)+2\sqrt{3\times 1}}$
= $\sqrt{3}+\sqrt{1}$
= $1+\sqrt{3}$
Jawaban: A

Soal No. 46

Jika $\sqrt{0,3+\sqrt{0,08}}=\sqrt{a}+\sqrt{b}$, maka $\frac{1}{a}+\frac{1}{b}$ = ….
A. 25
B. 20
C. 15
D. 10
E. 5

Penyelesaian: Lihat/Tutup

$\begin{align}\sqrt{0,3+\sqrt{0,08}} &= \sqrt{a}+\sqrt{b} \\ \sqrt{\frac{3}{10}+\sqrt{\frac{8}{100}}} &= \sqrt{a}+\sqrt{b} \\ \sqrt{\frac{3}{10}+\frac{\sqrt{8}}{10}} &= \sqrt{a}+\sqrt{b} \\ \sqrt{\frac{3+2\sqrt{2}}{10}} &= \sqrt{a}+\sqrt{b} \\ \frac{\sqrt{(2+1)+2\sqrt{2\times 1}}}{\sqrt{10}} &= \sqrt{a}+\sqrt{b} \\ \frac{\sqrt{2}+\sqrt{1}}{\sqrt{10}} &= \sqrt{a}+\sqrt{b} \\ \sqrt{\frac{2}{10}}+\sqrt{\frac{1}{10}} &= \sqrt{a}+\sqrt{b} \\ \sqrt{\frac{1}{5}}+\sqrt{\frac{1}{10}} &= \sqrt{a}+\sqrt{b} \end{align}$
Diperoleh $a=\frac{1}{5}$ dan $b=\frac{1}{10}$ maka:
$\frac{1}{a}+\frac{1}{b}=\frac{1}{\frac{1}{5}}+\frac{1}{\frac{1}{10}}=5+10=15$
Jawaban: C

Soal No. 47

Luas persegi panjang adalah $(6+2\sqrt{3})$ $\text{c}{{\text{m}}^{2}}$ dan lebarnya $(6-\sqrt{12})$ $\text{c}{{\text{m}}^{2}}$. Keliling persegi panjang adalah … cm.
A. $3+\sqrt{3}$
B. $4+4\sqrt{3}$
C. $16-4\sqrt{3}$
D. $8-2\sqrt{3}$
E. $16-2\sqrt{3}$

Penyelesaian: Lihat/Tutup

$l=6-\sqrt{12}=6-\sqrt{4\times 3}=6-2\sqrt{3}$
$\begin{align}L &= 6+2\sqrt{3} \\ p\times l &= 6+2\sqrt{3} \\ p(6-2\sqrt{3}) &= 6+2\sqrt{3} \\ p &= \frac{6+2\sqrt{3}}{6-2\sqrt{3}}\times \frac{6+2\sqrt{3}}{6+2\sqrt{3}} \\ p &= \frac{36+12\sqrt{3}+12\sqrt{3}+12}{36-12} \\ p &= \frac{48+24\sqrt{3}}{24} \\ p = 2+\sqrt{3} \end{align}$
Keliling persegi panjang adalah:
$\begin{align}K &= 2(p+l) \\ &= 2(2+\sqrt{3}+6-2\sqrt{3}) \\ &= 2(8-\sqrt{3}) \\ K &= 16-2\sqrt{3} \end{align}$
Jawaban: E

Soal No. 48

Sebuah daun jendela berbentuk persegi panjang dengan panjang $(20+4\sqrt{6})$ cm dan lebarnya $(20-4\sqrt{6})$ cm. Luas daun jendela tersebut adalah … $\text{c}{{\text{m}}^{2}}$.
A. 304
B. 496
C. $496-160\sqrt{2}$
D. $304+160\sqrt{2}$
E. $496+160\sqrt{2}$

Penyelesaian: Lihat/Tutup

$\begin{align}L &= p\times l \\ &= (20+4\sqrt{6})(20-4\sqrt{6}) \\ &= 400-96 \\ L &= 304 \end{align}$
Jadi, luas daun jendela itu adalah 304 $\text{c}{{\text{m}}^{2}}$.
Jawaban: A

Soal No. 49

Jika bilangan bulat $a$ dan $b$ memenuhi $\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}}=a\sqrt{6}+b$ maka $a+b$ = ….
A. $-6$
B. $-4$
C. $-3$
D. 3
E. 4

Penyelesaian: Lihat/Tutup

$\begin{align}\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}} &= a\sqrt{6}+b \\ \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}}\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}} &= a\sqrt{6}+b \\ \frac{2-\sqrt{6}-\sqrt{6}+3}{2-3}=a\sqrt{6}+b \\ \frac{5-2\sqrt{6}}{-1} &= a\sqrt{6}+b \\ 2\sqrt{6}-5 &= a\sqrt{6}+b \end{align}$
Diperoleh $a=2$ dan $b=-5$ maka:
$a+b=2+(-5)=-3$
Jawaban: C

Soal No. 50

$\frac{5}{2+\sqrt{3}}+\frac{4}{3+2\sqrt{3}}$ = ….
A. $6-\frac{7}{3}\sqrt{3}$
B. $6-\frac{1}{7}\sqrt{3}$
C. $6+\frac{1}{7}\sqrt{3}$
D. $6+\frac{7}{3}\sqrt{3}$
E. $6-\frac{3}{7}\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\frac{5}{2+\sqrt{3}}+\frac{4}{3+2\sqrt{3}}$
= $\frac{5}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}+\frac{4}{3+2\sqrt{3}}\times \frac{3-2\sqrt{3}}{3-2\sqrt{3}}$
= $\frac{10-5\sqrt{3}}{4-3}+\frac{12-8\sqrt{3}}{9-12}$
= $10-5\sqrt{3}+\frac{12-8\sqrt{3}}{-3}$
= $10-5\sqrt{3}-4+\frac{8\sqrt{3}}{3}$
= $6-\frac{15\sqrt{3}}{3}+\frac{8\sqrt{3}}{3}$
= $6-\frac{7}{3}\sqrt{3}$
Jawaban: A

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