Simplifying Algebraic Expressions

Here is the order of operations to follow:

1. Brackets

2. Exponents

3. Multiplication and division (from left to right)

4. Addition and subtraction (from left to right)

To remember the order, take the first letter of each step and make the acronym BEMDAS.

Simplify the following algebraic expression.

1. The two expressions inside brackets do not contain any like terms. So, they cannot be simplified any further.

2. Distribute the |-\dfrac{2}{5}| in front of the bracket by multiplying each of the terms inside the bracket by |-\dfrac{2}{5}|. ||\frac{1}{10}\color{blue} {-\dfrac{2}{5}}\times 3ab\color{blue} {-\dfrac{2}{5}}\times -\dfrac{3}{4}+(10a-8b)\div 2 \\ \dfrac{1}{10}-\dfrac{6}{5}ab+\dfrac{6}{20}+(10a-8b)\div 2||

3. Following the order of operations, divide. ||\frac{1}{10}-\frac{6}{5}ab+\frac{6}{20}\color{blue} {+10a\div 2 -8b\div 2}\\\frac{1}{10}-\frac{6}{5}ab+\frac{6}{20}+5a-4b||

4. Finally, simplify the like terms. To add |\dfrac{1}{10}| and |\dfrac{6}{20}|, find a common denominator. It is best to first simplify|\dfrac{6}{20}| by dividing the numerator and the denominator by |2.| ||\frac{6}{20}=\frac{6\color{blue} {\div 2}}{20\color{blue} {\div 2}}=\frac{3}{10}||
   
   The fractions now have a common denominator and can be simplified. ||\dfrac{1}{10}-\dfrac{6}{5}ab+\color{blue} {\dfrac{6}{20}}+5a-4b\\ \color{blue} {\dfrac{1}{10}}-\dfrac{6}{5}ab+\color{blue} {\dfrac{3}{10}}+5a-4b\\ \color{blue} {\dfrac{4}{10}}-\dfrac{6}{5}ab+5a-4b||

5. The fraction |\dfrac{4}{10}| can be simplified by dividing the numerator and denominator by |2.| ||\color{blue} {\frac{4}{10}}-\frac{6}{5}ab+5a-4b\\ \color{blue}{\frac{2}{5}}-\frac{6}{5}ab+5a-4b||

Answer: Arrange the terms in the expression alphabetically by decreasing order of degree. The simplified expression is |-\dfrac{6}{5}ab+5a-4b+\dfrac{2}{5}.|

Simplify the following algebraic expression. ||8(4x+12-5x)+8x^{3}\div2x^{2}||

1. Start by simplifying the like terms inside the brackets. Subtract |5x| from |4x|. ||\begin{align}8({\color{blue}{4x}}+12{\color{blue}{-5x}})&+8x^{3}\div2x^{2}\\ 8(-x+12)&+8x^{3}\div2x^{2}\end{align}||

2. Distribute the |8| in front of the brackets by multiplying each of the terms inside the bracket by |8|. ||\begin{align}\color{blue}{8\times}-x+\color{blue}{8\times}12&+8x^{3}\div2x^{2}\\ -8x + 96&+8x^3\div 2x^2\end{align}||

3. Following the order of operations, divide. ||\begin{align}-8x+96&+{\color{blue}{8x^{3}\div2x^{2}}}\\ -8x+96&+4x\end{align}||

4. Finally, simplify the like terms. Add |-8x| and |4x| together. ||\begin{align}\color{blue}{-8x}&+96\color{blue}{+4x}\\ -4x&+96\end{align}||

Answer: The simplified expression is |-4x+96.|

Simplify the following expression. ||6(x+3)-(3x^{3}+6x^{3}+8x^{2}-4x)+36x^{5}\div3x^{3}\times x+9||

1. Start by simplifying the like terms inside the brackets if possible. ||\begin{align} 6(x+3)-({\color{blue}{3x^{3}}}{\color{blue}{+6x^{3}}}+8x^{2}-4x)&+36x^{5}\div3x^{3}\times x+9\\ 6(x+3)-(9x^{3}+8x^{2}-4x)&+36x^{5}\div3x^{3}\times x+9\end{align}||

2. Distribute the |6| by multiplying each of the terms inside the first bracket by |6|. ||\begin{align}{\color{blue}{6\times x}}+{\color{blue}{6\times 3}}&-(9x^{3}+8x^{2}-4x)+36x^{5}\div3x^{3}\times x+9\\ 6x+18&-(9x^{3}+8x^{2}-4x)+36x^{5}\div3x^{3}\times x+9\end{align}||

3. Use the distributive property for the |-| in front of the second bracket. Remember that the negative sign means to multiply the terms inside the bracket by |-1|. So, multiply each term inside the second pair of brackets by |-1|. That is, reverse the signs. ||\begin{align}6x+18\color{blue}{-1\times 9x^{3}-1\times 8x^{2}-1\times -4x}&+36x^{5}\div3x^{3}\times x+9\\ 6x+18-9x^{3} - 8x^{2} + 4x& + 36x^{5}\div3x^{3}\times x+9\end{align}||

4. Divide, moving from left to right, where necessary. ||\begin{align}6x+18-9x^{3}-8x^{2}+4x&+{\color{blue}{36x^{5}\div3x^{3}}}\times x+9\\ 6x+18-9x^3-8x^2+4x&+12x^2\times x+9\end{align}||

5. Multiply, moving from left to right, where necessary. ||\begin{align}6x+18-9x^{3}-8x^{2}+4x&+{\color{blue}{12x^{2}\times x}}+9\\ 6x+18-9x^{3}-8x^{2}+4x&+12x^{3}+9\end{align}||

6. Add and subtract like terms. ||\color{blue}{6x}\color{fuchsia}{+18}\color{green}{-9x^{3}}-8x^{2}\color{blue}{+4x}\color{green}{+12x^{3}}\color{fuchsia}{+9}\\ 3x^{3}-8x^{2}+10x+27||

Answer: The simplified expression is |3x^{3}-8x^{2}+10x+27.|