Algebra: Adding and subtracting fractions
Simplifying fractions
A fraction stays the same if #\orange{\text{numerator}}# and #\blue{\text{denominator}}#: |
Examples |
1. are multiplied by the same number |
\[\begin{array}{rcll} \dfrac{\orange{3x+1}}{\blue{x+2}} &=& \dfrac{\orange{6x+2}}{\blue{2x+4}} \end{array}\] |
2. are multiplied by the same variable |
\[\begin{array}{rcll}\dfrac{\orange{x}}{\blue{y}} &=& \dfrac{\orange{x \cdot z}}{\blue{y \cdot z}} \end{array}\] |
3. are divided by the same number |
\[\begin{array}{rcll} \dfrac{\orange{4x+2}}{\blue{2x+2}} &=& \dfrac{\orange{2x+1}}{\blue{x+1}}\end{array}\] |
4. are divided by the same variable |
\[\begin{array}{rcll}\dfrac{\orange{x}}{\blue{x^2+x}} &=& \dfrac{\orange{1}}{\blue{x+1}} \end{array}\] |
The process of making the numerator and denominator smaller is called the simplification of an expression. |
Example \[\dfrac{\orange{x}}{\blue{x^2+x}} = \dfrac{\orange{1}}{\blue{x+1}} \] |
#\begin{array}{rcl}
\dfrac{9\cdot a^3\cdot b^2\cdot \left(c+1\right)^4}{a^6\cdot b^6\cdot \left(c+1\right)^6}
&=& \displaystyle {{9}\over{a^3\cdot b^4\cdot \left(c+1\right)^2}}
\\ && \phantom{xxx}\blue{\text{common factor divided by } a^3\cdot b^2\cdot \left(c+1\right)^4 \text{ }}
\end{array}#
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