 |
| a. | $\begin{align}& (^3\sqrt{8})^2\times{\frac{1}{^3\sqrt27}}\\[5pt]
= &(8^{\frac{1}{3}})^2\times{\frac{1}{27^\frac{1}{3}}}\\[5pt]
=&({2^{{3}\times{\frac{1}{3}}}})^2\times{\frac{1}{{3^{{3}\times\frac{1}{3}}}}}\\[5pt]
=&{2^2}\times{\frac{1}{3}}\\[5pt]
=&\underline{\underline{\frac{4}{3}}}\\[5pt]
\end{align}$ | b. | $\begin{align}& (\sqrt{125})^3\times{\frac{1}{\sqrt20}}\times 10\\[5pt]
=& (\sqrt{5^3})^3\times{\frac{10}{\sqrt20}}\\[5pt]
=&({5^{{3}\times{\frac{1}{2}}}})^3\times{\sqrt\frac{100}{20}}\\[5pt]
=&{5^{\frac{9}{2}}\times{\sqrt{5}}}\\[5pt]
=&{5^{\frac{9}{2}}\times{{5^\frac{1}{2}}}}\\[5pt]
=&{5^{{\frac{9}{2}}+{\frac{1}{2}}}}\\[5pt]
=&{5^5}\\[5pt]
&= \underline{\underline{3125}}\\[5pt]
\end{align}$ |
| c. | $\begin{align}&\frac{ {32^{-\frac{2}{5}}}\times216^{\frac{2}{3}}}{{81^\frac{3}{4}\times^3\sqrt8^0}\times^3\sqrt27^{-2}}\\[5pt]
=&\frac{{ ({2^5})^{-\frac{2}{5}}}\times{{6}^{3\times\frac{2}{3}}}}{{3}^{4\times\frac{3}{4}}\times^3\sqrt{1}\times[(3^3)^{-2}]^\frac{1}{3}}\\[5pt]
=&\frac{2^{-2}\times6^2}{3^3\times1\times3^{-2}}\\[5pt]
=&{\frac{\frac{1}{4}\times36} {3^1}}\\[5pt]
=&{\frac{9}{3}}\\[5pt]
=&\underline{\underline{3}}\\[5pt]
\end{align}$ | d. | $\begin{align}& {\sqrt\frac{18\times5^2}{8}}\\[5pt]
=& {\sqrt\frac{9\times5^2}{4}}\\[5pt]
=& ({\frac{3^2\times5^2}{2^2}})^\frac{1}{2}\\[5pt]
=& ({\frac{3\times5}{2}})^{2\times\frac{1}{2}}\\[5pt]
=&\frac{15}{2}\\[5pt]
=&\underline{\underline{7\frac{1}{2}}}\\[5pt]
\end{align}$
|
| e. | $\begin{align}& ({\frac{1}{8})^{-\frac{1}{3}} \times{5^{-2}}\times100}\\[5pt]
=& 8^\frac{1}{3}\times{\frac{1}{5^2}} \times100\\[5pt]
&{2^{3\times{\frac{1}{3}}}}\times{\frac{1}{25}}\times 100\\[5pt]
=&2\times4\\[5pt]
=&\underline{\underline{8}}\\[5pt]
\end{align}$ | f. | $\begin{align}& 27^{\frac{2}{3}}-16^{\frac{3}{4}}\\[5pt]
=&{3^{3\times{\frac{2}{3}}}}-{2^{4{\times{^\frac{3}{4}}}}}\\[5pt]
=&{3^2}-{2^3}\\[5pt]
=&9-8\\[5pt]
=& \underline{\underline{1}}\\[5pt]
\end{align}$
|
 |
| a. | $\begin{align}& \sqrt{ a{^2}b^{-\frac{1}{2}}}\\[5pt]
=& (a^2b^{-\frac{1}{2}})^\frac{1}{2}\\[5pt]
=& a^{2\times\frac{1}{2}}b^{{-\frac{1}{2}}\times\frac{1}{2}}\\[5pt]
=& a\times b^{-\frac{1}{4}}\\[5pt]
=&\underline{\underline{\frac{a}{b\frac{1}{4}}}}\\[5pt]
\end{align}$ | b. | $\begin{align}& ({x}^{-4})^\frac{1}{2}\times{\frac{1}{\sqrt {x^{-3}}}}\\[5pt]
=&{x^{-4\times{\frac{1}{2}}}}\times\frac{1}{x^{-3\times{\frac{1}{2}}}}\\[5pt]
= &{x}^{-2}\times{\frac{1}{x^{-\frac{3}{2}}}}\\[5pt]
= &{x}^{-2}\times{x^\frac{3}{2}}\\[5pt]
= &{x^{-2\times{\frac{3}{2}}}}\\[5pt]
= & x^{-\frac{1}{2}}\\[5pt]
=&\underline{\underline{\frac{1}{x^\frac{1}{2}}}}\\[5pt]
\end{align}$
|
| c. | $\begin{align}& ({x}^\frac{1}{2}-{x^{{-\frac{1}{2}}}}) ({x}^\frac{1}{2}+{x^{{-\frac{1}{2}}}})\\[5pt]
=& ({x}^\frac{1}{2})^2-({x}^{-\frac{1}{2}})^2\\[5pt]
=&x{^{\frac{1}{2}\times 2}}-x{^{-\frac{1}{2}\times 2}}\\[5pt]
=& x-{x}^{-1}\\[5pt]
=& x-\frac{1}{x}\\[5pt]
=&\underline{\underline{\frac{x^2-1}{x}}}\\[5pt]
\end{align}$
| d. | $\begin{align}& (x\div n\sqrt{x})^n\\[5pt]
=& (x^1\div {x}^\frac{1}{n})^n\\[5pt]
=& {x^ n}\div x{^{\frac{1}{n}\times n}}\\[5pt]
=& {x ^n}\div {x}^{1}\\[5pt]
=&\underline{\underline{x}^{n-1}}\\[5pt]
\end{align}$ |
| e. | $\begin{align}& [(\sqrt a^3)^{-2}]^\frac{1}{2}\\[5pt]
& [((a^3)^\frac{1}{2})^{-2}]^\frac{1}{2}\\[5pt]
= &{a}^{-\frac{3}{2}}\\[5pt]
=&\underline{\underline{\frac{1}{a\frac{3}{2}}}}\\[5pt]
\end{align}$ | | |
 |
| a. | $\begin{align}&lg(\frac {217}{38}\div \frac {31}{266})=2\lg 7\\[5pt]
&lg(\frac {217}{38}\times \frac {266}{31})=2\lg 7\\[5pt]
&lg(\frac {7}{1}\times \frac {7}{1})=2\lg 7\\[5pt]
&\lg 7^2=\lg 7^2\\[5pt]
&2\lg 7=2\lg 7\\[5pt]
\end{align}$ | b. | $\begin{align}&\frac {1}{2}\lg 9 +\lg2=2\lg 3-\lg 1.5\\[5pt]
&\lg 9^\frac {1}{2}+\lg2=\lg 3^2-\lg 1.5\\[5pt]
&\lg{3^{2\times{\frac{1}{2}}}}+\lg2=\lg\frac {9}{1.5}\\[5pt]
&\lg 3+\lg2=\lg\frac {9}{1.5}\\[5pt]
&\lg 6=\lg 6\\[5pt]
\end{align}$ |
| c. | $\begin{align}&\lg_3 24 +\lg_3 5-\lg_3 40=1\\[5pt]
&\lg_3 \frac{24\times 5}{40}=1\\[5pt]
&\lg_3 3=1\\[5pt]
&1=1\\[5pt]
\end{align}$ | d. | $\begin{align}&\lg 26 +\lg 119-\lg 51-\lg91=\lg2-\lg3\\[5pt]
&\lg\frac{26\times 119}{51\times91}=\lg2-\lg3\\[5pt]
&\lg\frac{2}{3}=\lg\frac{2}{3}\\[5pt]
\end{align}$ |
| e. | $\begin{align}&2\lg_a 3 +\lg_a 20-\lg_a36=\lg_a 10-\lg_a 2\\[5pt]
&\lg_a 3^2 +\lg_a 20-\lg_a36=\lg_a\frac {10}{2}\\[5pt]
&\lg_a\frac{9\times 20}{36}=\lg _a 5\\[5pt]
&\lg_a 5=\lg _a 5\\[5pt]
\end{align}$ | | |
By studying this lesson, you will be able to
Simplify expressions involving powers and roots and solve equations using the laws of indices and logarithms.