সূচক ও লগারিদম ( সরল কর ) অনু:৪.১ গানিতীক সমাধান- ১

সরল কর :

1) \displaystyle \frac{7^{3} X7^{-3}}{3X3^{-4}}

2) \displaystyle \frac{\sqrt[3]{7^{2}} .\sqrt[3]{7}}{\sqrt{7}}

3) \displaystyle \left( 2^{-1} +5^{-1}\right)^{-1}

4) \displaystyle \left( 2a^{-1} +3b^{-1}\right)^{-1}

5) \displaystyle \left(\frac{a^{2} b^{-1}}{a^{-2} b}\right)^{2}

1) সমাধান,,,

\displaystyle \frac{7^{3} X7^{-3}}{3X3^{-4}}

\displaystyle =\frac{7^{3+( -3)}}{3^{1+( -4)}}

\displaystyle =\frac{7^{3-3}}{3^{1-4}}

\displaystyle =\frac{7^{0}}{3^{-3}}

\displaystyle =\frac{1}{3^{-3}}

\displaystyle =1X3^{3}

\displaystyle =1X3X3X3

\displaystyle =27

2) সমাধান,,

\displaystyle \frac{\sqrt[3]{7^{2}} .\sqrt[3]{7}}{\sqrt{7}}

\displaystyle =\frac{\left( 7^{2}\right)^{\frac{1}{3}} .7^{\frac{1}{3}}}{7^{\frac{1}{2}}}

\displaystyle =\frac{7^{\frac{2}{3} +\frac{1}{3}}}{7^{\frac{1}{2}}}

\displaystyle =7^{\frac{2}{3} +\frac{1}{3} -\frac{1}{2}}

\displaystyle =7^{\frac{4+2-3}{6}}

\displaystyle =7^{\frac{3}{6}}

\displaystyle =7^{\frac{1}{2}}

\displaystyle =\sqrt{7}

3) সমাধান,,

\displaystyle \left( 2^{-1} +5^{-1}\right)^{-1}

\displaystyle =\left(\frac{1}{2} +\frac{1}{5}\right)^{-1}

\displaystyle =\left(\frac{5+2}{10}\right)^{-1}

\displaystyle =\left(\frac{7}{10}\right)^{-1}

\displaystyle =\frac{1}{\frac{7}{10}}

\displaystyle =1X\frac{10}{7}

\displaystyle =\frac{10}{7} (Ans):

4) সমাধান,,,

\displaystyle \left( 2a^{-1} +3b^{-1}\right)^{-1}

\displaystyle =\left( 2X\frac{1}{a} +3X\frac{1}{b}\right)^{-1}

\displaystyle =\left(\frac{2}{a} +\frac{3}{b}\right)^{-1}

\displaystyle =\left(\frac{2b+3a}{ab}\right)^{-1}

\displaystyle =\frac{1}{\frac{2b+3a}{ab}}

\displaystyle =1X\frac{ab}{2b+3a}

\displaystyle =\frac{ab}{3a+2b}

5) সমাধান,,

\displaystyle \left(\frac{a^{2} b^{-1}}{a^{-2} b}\right)^{2}

\displaystyle =\left( a^{2+2} .b^{-1-1}\right)^{2}

\displaystyle =\left( a^{4} .b^{-2}\right)^{2}

\displaystyle =a^{8} b^{-4}

\displaystyle =a^{8} .\frac{1}{a^{4}}

\displaystyle =\frac{a^{8}}{a^{4}}

1) \displaystyle \sqrt{x^{-1} y} .\sqrt{y^{-1} z} .\sqrt{z^{-1} x}
\displaystyle ( x >0,y >0,z >0)

2) \displaystyle \frac{2^{n+4} -4.2^{n+1}}{2^{n+2} \div 2}
3) \displaystyle \frac{3^{m+1}}{\left( 3^{m}\right)^{m-1}} \div \frac{9^{m+1}}{\left( 3^{m-1}\right)^{m+1}}

1) সমাধান,,

\displaystyle \sqrt{x^{-1} y} .\sqrt{y^{-1} z} .\sqrt{z^{-1} x}

\displaystyle =\left( x^{-1} y\right)^{\frac{1}{2}} .\left( y^{-1} z\right)^{\frac{1}{2}} .\left( z^{-1} x\right)^{\frac{1}{2}}

\displaystyle =\left( x^{-1} y.y^{-1} z.z^{-1} x\right)^{\frac{1}{2}}

\displaystyle =\left( \ x^{-1+1} .y^{1-1} .z^{1-1}\right)^{\frac{1}{2}}

\displaystyle =\left( x^{0} .y^{0} .z^{0}\right)^{\frac{1}{2}}

\displaystyle =( 1X1X1)^{\frac{1}{2}}

\displaystyle =1

1) বিকল্প সমাধান,,

\displaystyle \sqrt{x^{-1} y} .\sqrt{y^{-1} z} .\sqrt{z^{-1} x}

\displaystyle =\sqrt{\frac{1}{x} .y} \ \sqrt{\frac{1}{y} .z} \ \ \sqrt{\frac{1}{z} .x}

\displaystyle =\sqrt{\frac{y}{x}} .\sqrt{\frac{z}{y} \ } .\sqrt{\frac{x}{z}}

\displaystyle =\frac{\sqrt{y}}{\sqrt{x}} .\frac{\sqrt{z}}{\sqrt{y}} .\frac{\sqrt{x}}{\sqrt{z}}

\displaystyle =1

2) সমাধান,,

\displaystyle \frac{2^{n+4} -4.2^{n+1}}{2^{n+2} \div 2}

\displaystyle =\frac{2^{n} .2^{4} -4X2^{n} .2^{1}}{2^{n+2} \div 2^{1}}

\displaystyle =\frac{2^{n} .16-8.2^{n}}{2^{n+2-1}}

\displaystyle =\frac{2^{n}( 16-8)}{2^{n+1}}

\displaystyle =\frac{2^{n} X8}{2^{n} X2^{1}}

\displaystyle =\frac{8}{2}

\displaystyle =4 ( Ans ):

3) সমাধান,,,

\displaystyle \frac{3^{m+1}}{\left( 3^{m}\right)^{m-1}} \div \frac{9^{m+1}}{\left( 3^{m-1}\right)^{m+1}}

\displaystyle =\frac{3^{m+1}}{3^{m^{2} -m}} \div \frac{\left( 3^{2}\right)^{m+1}}{3^{( m-1)( m+1)}}

\displaystyle =\frac{3^{m+1}}{3^{m^{2} -m}} \div \frac{3^{2m+2}}{3^{m^{2} -1}}

\displaystyle =3^{m+1-m^{2} +m} \div 3^{2m+2-m^{2} +1}

\displaystyle =3^{2m-m^{2} +1} \div 3^{2m-m^{2} +3}

\displaystyle =3^{\left( 2m-m^{2} +1\right) -\left( 2m-m^{2} +3\right)}

\displaystyle =3^{2m-m^{2} +1-2m+m^{2} -3}

\displaystyle =3^{-2}

\displaystyle =\frac{1}{3^{2}}

\displaystyle =\frac{1}{3X3}

\displaystyle =\frac{1}{9} ( Ans):

সরল কর : কাজ: –

\displaystyle i) \ \frac{2^{4} .2^{2}}{32}

\displaystyle ii) \ \left(\frac{2}{3}\right)^{\frac{5}{2}} X\left(\frac{2}{3}\right)^{-\frac{5}{2}}

\displaystyle iii) \ 8^{\frac{3}{4}} \div 8^{\frac{1}{2}}

i) সমাধান,,


\displaystyle i) \ \frac{2^{4} .2^{2}}{32}
\displaystyle =\frac{2^{4+2}}{2X2X2X2X2} [ we know that,, \displaystyle a^{m} Xa^{n} =a^{m+n} ]
\displaystyle =\frac{2^{6}}{2^{5}}
\displaystyle =2^{6-5}
\displaystyle =2^{1}
\displaystyle =2 ( Ans:)

ii) সমাধান,,

\displaystyle ii) \ \left(\frac{2}{3}\right)^{\frac{5}{2}} X\left(\frac{2}{3}\right)^{-\frac{5}{2}}

\displaystyle =\left(\frac{2}{3}\right)^{\frac{5}{2} -\frac{5}{2}}

\displaystyle =\left(\frac{2}{3}\right)^{0}

\displaystyle =1 ( Ans:)

iii) সমাধান,,

\displaystyle iii) \ 8^{\frac{3}{4}} \div 8^{\frac{1}{2}}

\displaystyle =8^{\frac{3}{4} \ -\ \frac{1}{2}} [ We know that,, \displaystyle a^{m} \div a^{n} =a^{m-n} ]

\displaystyle =8^{\frac{3-2}{4}}

\displaystyle =8^{\frac{1}{4}}

\displaystyle =( 2X2X2)^{\frac{1}{4}}

\displaystyle =\left( 2^{3}\right)^{\frac{1}{4}}

\displaystyle =2^{\frac{3}{4}} (Ans:)

Post Author: showrob

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