Menyederhanakan Bentuk Akar dan Pembahasan Soal

Untuk menyederhanakan atau menjabarkan bentuk akar. terlebih dahulu kita harus memahami sifat-sifat berikut ini.

Teorema:

1. $\sqrt[n]{a^n} = a$, jika n ∊ ganjil

2. $\sqrt[n]{a^n}=|a|=\left\{\begin{matrix}a, & bila \ a \geq 0 \ dan \ n \ genap \\ -a, & bila \ a <  0 \ dan \ n \ genap\end{matrix}\right.$
3. $\sqrt[n]{0} = 0$
4. $\sqrt[n]{a^{mn+p}b^q} = a^m \sqrt[n]{a^pb^q}$, p < n dan q > n
5. $\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$
6. $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
7. $\sqrt[m]{\sqrt[n]{a}} = \sqrt[n]{\sqrt[m]{a}}$
8. $\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$
9. $\sqrt[m]{a^n} = (\sqrt[m]{a})^n$
10. $\sqrt[np]{a^{mp}} = \sqrt[n]{a^m}$

Contoh Soal 1

Sederhanakan atau jabarkanlah! 

a. $\sqrt{72}$ b. $\sqrt[3]{1.296}$

c. $\sqrt[4]{576a^{17} b^8 c^9}$ d. $\sqrt[n]{a^{10n+3}}$

 Jawab:

a. $\sqrt{72}$

= $\sqrt{72}= \sqrt{36.2} = \sqrt{6^2. 2}= 6 \sqrt{2}$

b. $\sqrt[3]{1.296}= \sqrt[3]{2^4 \times 3^4}$

= $2 \times 3 \sqrt[3]{2 \times 3}$

= $2 \times \sqrt[3]{2} \times 3 \times \sqrt[3]{3}= 6 \sqrt[3]{6}$

c. $\sqrt[4]{576a^{17} b^8 c^9}= \sqrt[4]{2^6 3^2 a^{17} b^8 c^9}$

= $\sqrt[4]{2^4 a^{16} b^8 c^8} \times \sqrt[4]{2^2 3^2 ac}$

= $2a^4c^2 \sqrt[4]{36ac}$

d. $\sqrt[n]{a^{10n+3}} = \sqrt[n]{a^{10n}. a^3}$

= $\sqrt[n]{a^{10n}} \times \sqrt[n]{a^3}=a^{10} \sqrt[n]{a^3}$

Contoh Soal 2

Sederhanakanlah!

a. $\sqrt{x^2 +2x+1}$

b. $a+2b+ \sqrt{a^2 -4ab +4b^2}$, untuk a = 2 dan b = 4.

Jawab:

a. $\sqrt{x^2 +2x+1}= \sqrt{(x+1)^2}=|x+1|$

b. $a+2b+ \sqrt{a^2 -4ab +4b^2}$

= $a+2b+ \sqrt{(a-2b)^2}$

= $a+2b+ |a-2b|$, jika a ≤ 2b maka

= $a+2b- (a-2b)$

= $a+2b-a+2b=4b = 16$

Contoh Soal 3

Ubahlah ke dalam bentuk radikal penuh, yaitu radikal dengan koefisien 1.

a. $3\sqrt{5}$ b. $4x^2 \sqrt[3]{y^2}$

c. $\frac{x-y}{x+y} \sqrt{\frac{x+y}{x-y}}$

Jawab:

a. $3\sqrt{5}=  \sqrt{3^2 \times 5} = \sqrt{45}$

b. $4x^2 \sqrt[3]{y^2} = \sqrt[3]{(4x^2)^3 y^2} = \sqrt[3]{64x^6y^2}$

c. $\frac{x-y}{x+y} \sqrt{\frac{x+y}{x-y}}$

= $\sqrt{\left( \frac{x-y}{x+y} \right)^2 \times \frac{x+y}{x-y}}$

= $\sqrt{ \frac{(x-y)^2}{(x+y)^2} \times \frac{x+y}{x-y}}$

= $\sqrt{\frac{x-y}{x+y}}$

Contoh Soal 4

Sederhanakan atau jabarkanlah.

a. $\sqrt{\frac{5}{6}}$ b. $20 \sqrt{\frac{19}{50}}$ c. $\sqrt[12]{\frac{c^7}{a^7b^{10}}}$ d. $\frac{1}{\sqrt[3]{x^2y^5}}$

e. $\sqrt[3]{\sqrt{4.096x^{12}y^{27}}}$ f. $\sqrt[5]{\sqrt[3]{5^{30}a^{40}b^{25}}}$ g. $\sqrt[8]{x^2y^6}$ h. $\sqrt[15]{32a^{10}b^5c^{25}}$

Jawab:

a. $\sqrt{\frac{5}{6}} = \sqrt{\frac{5}{6} \times \frac{6}{6}} = \sqrt{\frac{30}{6^2}}$

= $\frac{\sqrt{30}}{\sqrt{6^2}} = \frac{1}{6} \sqrt{30}$

b. $20 \sqrt{\frac{19}{50}}=20 \sqrt{\frac{19}{50} \times \frac{2}{2}} = 20 \sqrt{\frac{38}{100}}$

= $20 \frac{\sqrt{38}}{\sqrt{100}}=20 \frac{\sqrt{38}}{10} = 2 \sqrt{38}$

c. $\sqrt[12]{\frac{c^7}{a^7b^{10}}}= \sqrt[12]{\frac{c^7}{a^7b^{10}} \times \frac{a^5b^2}{a^5b^2}}$

= $\sqrt[12]{\frac{a^5b^2c^7}{a^{12}b^{12}}} = \frac{\sqrt[12]{a^5b^2c^7}}{\sqrt[12]{a^{12}b^{12}}}$

= $\frac{\sqrt[12]{a^5b^2c^7}}{ab}= \frac{1}{ab} \sqrt[12]{a^5b^2c^7}$

d. $\frac{1}{\sqrt[3]{x^2y^5}} = \frac{1}{\sqrt[3]{x^2y^5}} \times \frac{\sqrt[3]{xy}}{\sqrt[3]{xy}}$

= $\frac{\sqrt[3]{xy}}{\sqrt[3]{x^3y^6}} = \frac{\sqrt[3]{xy}}{xy^2}= \frac{1}{xy^2} \sqrt[3]{xy}$

e. $\sqrt[3]{\sqrt{4.096x^{12}y^{27}}} = \sqrt[6]{2^{12}x^{12}y^{27}}$

= $2^2x^2y^4 \sqrt[6]{y^3}=4x^2 y^4 \sqrt{y}$

f. $\sqrt[5]{\sqrt[3]{5^{30}a^{40}b^{25}}}= \sqrt[15]{5^{30}a^{40}b^{25}}$

= $5^2a^2b \sqrt[15]{a^{10}b^{10}}=25a^2b \sqrt[15]{a^{10}b^{10}}$

g. $\sqrt[8]{x^2y^6} = \sqrt[4]{xy^3}$

h. $\sqrt[15]{32a^{10}b^5c^{25}}= \sqrt[15]{2^5 a^{10}b^5 c^{25}}$

= $\sqrt[3]{2a^2bc^5} = c\sqrt[3]{2a^2bc^2}$