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EKSPONEN

Sifat-sifat eksponen:

1. $a^{m}\: \cdot \: a^{n}\: =\: a^{m+n}$

2. $\frac{a^{m}}{a^{n}}\: =\:a^{m-n}$

3. $\left ( a^{m} \right )^{n}\: =\: a^{m\cdot n}$

4.   $\left ( a^{m}\cdot a^{n} \right )^{p}\: =\: a^{mp}\cdot a^{np}$

5.    $a^{0}\: =\: 1$

6.    $a^{-n}\: =\: \frac{1}{a^{n}}$

7. $\frac{1}{a^{-n}}\: =\: a^{n}$

8. $a^{\frac{m}{n}}\: =\: \sqrt[n]{a^{m}}$

Soal dan Pembahasan:

1. UM-UGM 2006 Kode 381
   Bentuk sederhana dari
   $\frac{\left ( x^{-4}y^{\frac{2}{3}} \right )^{-\frac{1}{2}}\left ( x^{-\frac{7}{3}}y^{-1} \right )^{\frac{1}{2}}}{\left ( x^{\frac{1}{2}}y^{3} \right )^{-\frac{1}{6}}\left ( x^{-\frac{1}{4}}y^{-1} \right )^{\frac{1}{3}}}$
   adalah .....
   A.

   D. $\frac{x}{y}$
   E. $\frac{y}{x}$

Pembahasan:

$\frac{\left ( x^{-4}y^{\frac{2}{3}} \right )^{-\frac{1}{2}}\left ( x^{-\frac{7}{3}}y^{-1} \right )^{\frac{1}{2}}}{\left ( x^{\frac{1}{2}}y^{3} \right )^{-\frac{1}{6}}\left ( x^{-\frac{1}{4}]y^{-1} \right )^{\frac{1}{3}}}$
Pergunakan sifat eksponen No. 4

$\frac{\left ( x^{-4\: \cdot \: -\frac{1}{2}}\: y^{\frac{2}{3}\: \cdot \:- \frac{1}{2}} \right )\left ( x^{-\frac{7}{3}\: \cdot \: \frac{1}{2}}\: y^{-1\: \cdot \: \frac{1}{2}} \right )}{\left ( x^{\frac{1}{2}\: \cdot \: -\frac{1}{6}}\: y^{3\: \cdot \: -\frac{1}{6}} \right )\left ( x^{-\frac{1}{4}\: \cdot \: \frac{1}{3}} \: y^{-1\: \cdot \: \frac{1}{3}}\right )}$

$=\frac{\left (x^{2}\: y^{-\frac{1}{3}} \right )\left ( x^{-\frac{7}{6}}\: y^{-\frac{1}{2}} \right )}{\left (x^{-\frac{1}{12}}\: y^{-\frac{1}{2}} \right )\left ( x^{-\frac{1}{12}}\: y^{-\frac{1}{3}} \right )}$

$=\: x^{2-\frac{7}{6}+\frac{1}{12}+\frac{1}{12}}\: y^{-\frac{1}{3}-\frac{1}{2}+\frac{1}{2}+\frac{1}{3}}$
$=x\: y^{0}$
$=\: x$

Jawaban _____________________ (B)

2. SIMAK UI 2012 Kode 222

   $\frac{5^{4022}-5^{4018}}{5^{4020}-5^{4016}}=\: .....$
   A.

   C. $\frac{25}{4}$
   D. $\frac{25}{2}$
   E.

Pembahasan:

$\frac{5^{4022}-5^{4018}}{5^{4020}-5^{4016}}=\:$ $\frac{5^{4018}\left ( 5^{4}-1 \right )}{5^{4016}\left ( 5\: ^{4} -1\right )}$
$=\frac{5^{4018}}{5^{4016}}$
   $=5^{4018-4016}$
   $=5^{2}$

Jawaban _____________________ (E)

3. SNMPTN 2008 Kode 201

Dalam bentuk pangkat positif, $\frac{x^{-2}-y^{-2}}{\left ( xy \right )^{-2}}=\: .....$
A. $\left ( x-y \right )\left ( x-y \right )$
B. $-\left ( x+y \right )\left ( x-y \right )$
C. $\left ( x-y \right )^{2}$
D. $x\left ( x-y \right )$
E. $-x\left ( x-y \right )$

Pembahasan:

   $\frac{x^{-2}-y^{-2}}{\left ( xy \right )^{-2}}=\:$ $\frac{\frac{1}{x^{2}}-\frac{1}{y^{2}}}{\frac{1}{\left ( xy \right )^{2}}}$
$=\frac{\frac{y^{2}-x^{2}}{x^{2}y^{2}}}{\frac{1}{\left ( xy \right )^{2}}}$
$=\frac{y^{2}-x^{2}}{\left ( xy \right)^2}\: \cdot \: \left ( xy \right )^{2}$
    $=y^{2}-x^{2}$
      $=-x^{2}+y^{2}$
    $=-\left ( x^{2} -y^{2}\right )$
$=-\left ( x+y \right )\left ( x-y \right )$

Jawaban _____________________ (B)

4. SNMPTN 2008 Kode 301

Dalam bentuk pangkat rasional $\sqrt[3]{x^{3}\sqrt[5]{x^{3}\sqrt{x^{3}}}}=\: .....$

A. $x^{\frac{13}{30}}$
B. $x^{\frac{3}{30}}$
C. $x^{\frac{13}{10}}$
D. $x^{\frac{31}{10}}$
E. $x^{\frac{39}{10}}$

Pembahasan:

$\sqrt[3]{x^{3}\sqrt[5]{x^{3}\sqrt{x^{3}}}}=\:$ $\sqrt[3]{x^{3}\sqrt[5]{x^{3}\left ( x^{\frac{3}{2}} \right )}}$
   $=\sqrt[3]{x^{3}\sqrt[5]{x^{3+\frac{3}{2}}}}$
$=\sqrt[3]{x^{3}\sqrt[5]{x^{\frac{9}{2}}}}$
                        $=\sqrt[3]{x^{3}\left ( x^{\frac{9}{2}} \right )^{\frac{1}{5}}}$
$=\sqrt[3]{x^{3}\left ( x^{\frac{9}{10}} \right )}$
   $=\sqrt[3]{x^{3+\frac{9}{10}}}$
$=\sqrt[3]{x^{\frac{39}{10}}}$
$=\left ( x^{\frac{39}{10}} \right )^{\frac{1}{3}}$
$=x^{\frac{13}{10}}$

Jawaban _____________________ (C)

$\sqrt[p]{x^{m}\sqrt[q]{x^{n}\sqrt[r]{x^{o}}}}=\: x^{\frac{mqr+nr+o}{pqr}}$
$\sqrt[3]{x^{3}\sqrt[5]{x^{3}\: \sqrt[]{x^{3}}}}=\: x^{\frac{3\cdot 5\cdot 2+3\cdot 2+3}{3\cdot 5\cdot 2}}$
   $=x^{\frac{39}{30}}$
   $=x^{\frac{13}{10}}$

5. USM STIS 2012

Bentuk sederhana dari $\frac{y^{-1}+xy^{-2}}{1-x^{2}y^{-2}}$ adalah .....
    A. $\frac{1}{y-x}$
B. $\frac{1}{y+x}$
C. $\frac{1}{x-y}$
D.

Pembahasan:

$\frac{y^{-1}+xy^{-2}}{1-x^{2}y^{-2}}\: =\: \frac{\frac{1}{y}+\frac{x}{y^{2}}}{1-\frac{x^{2}}{y^{2}}}$
   $=\frac{\frac{y+x}{y^{2}}}{\frac{y^{2}-x^{2}}{y^{2}}}$
   $=\frac{y+x}{y^{2}-x^{2}}$
   $=\frac{y+x}{\left ( y+x \right )\left ( y-x \right )}$
   $=\frac{1}{y-x}$

Jawaban _____________________ (A)

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Senin, 07 Agustus 2017

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