Simplifying Algebraic Expressions

To simplify algebraic expressions using addition and subtraction, recognise the like terms, then add or subtract accordingly.

Examples:

$7x-x=6x$

${ 2x }^{ 2 }+{ 5x }^{ 2 }={ 7x }^{ 2 }$

$3a-4b+5a-b\\ =\quad 3a+5a-4b-b\\ =\quad 8a-5b$

$x^{ 2 }+3xy-8y^{ 3 }+5x^{ 2 }-yx\\ =\quad 6x^2+2xy-8y^3$

Multiplication

To multiply algebraic expressions, multiply numbers with numbers and letters with corresponding letters. Powers of the terms you multiply are added, hence:

${ a }^{ m }\times { a }^{ n }={ a }^{ m+n }$

Examples:

$5x \times 3 = 15x$

$-3a \times 5b \times 2c =-30abc$

${ 3 }x^{ 3 }\times 4{ x }^{ 2 }{ y }^{ 3 }\\ =\quad (3\times 4)\times { x }^{ 3+2 }\times { y }^{ 3 }\\ =\quad 12{ x }^{ 5 }{ y }^{ 3 }$

$2{ p }^{ \frac { 3 }{ 2 } }{ q }^{ \frac { 1 }{ 2 } }\times 7{ p }^{ \frac { 3 }{ 4 } }{ q }^{ 2 }\\ =\quad (2\times 7)\times { p }^{ \frac { 3 }{ 2 } +\frac { 3 }{ 4 } }\times { q }^{ \frac { 1 }{ 2 } +2 }\\ =\quad 14{ p }^{ \frac { 9 }{ 4 } }{ q }^{ \frac { 5 }{ 2 } }$

Division

In division, the powers of corresponding like terms are subtracted. Notice that this is the opposite of multiplication.

${ a }^{ m }\div { a }^{ n }={ a }^{ m-n }$

Examples:

${ a }^{ 4 }\div { a }^{ 2 }\\ = { a }^{ 4-2 }\\= { a }^{ 2 }$

Using cancelling or index laws to simplify divisions makes it easier. Cancel one variable on the top with another similar variable on the bottom.

${ 6 }v^{ 2 }y\div 2vy\\ = \quad \cfrac { 6\times \not v \times v\times \not y }{ 2\times \not v \times \not y } \\ =\quad 3v$

$\cfrac { 5{ a }^{ 3 }b }{ 15{ a }b^{ 2 } } = \cfrac { 5\times \not a \times a\times a\times \not b }{ 15\times \not a \times \not b \times b } = \cfrac { { a }^{ 2 } }{ 3b }$

Mixture of Algebraic Operations

Different operations could be done separately in its proper order. BODMAS operations is definitely useful to remember, as well as a good knowledge of Index Laws.

BODMAS is an abbreviation of

1. Brackets

Any operations governed inside brackets take the highest priority. Brackets could come in many forms; e.g. parentheses, braces, square brackets; and the most inner brackets in an algebraic operation (or any function) is to be completed first.

2. Order

Order refers to indices, powers or exponents - these take the second priority.

3. Division/Multiplication

Then comes division and multiplication. These two take the next level of priority, however none is greater than the other. In a line of division and multiplication, complete the left first, then work your way through to the right.

The last priority are either addition or subtraction. Similar to division/multiplication, there is none greater than the other, start solving operations from the left to the right.

Examples

1. ${ a }^{ 2 }\times \left( 14ab-5ba \right) \div 3{ a }^{ 2 }b$

$=\quad { a }^{ 2 }\times 9ab\div 3{ a }^{ 2 }b\\ =\quad \cfrac { { a }^{ 2 }\times 9ab }{ 3{ a }^{ 2 }b } \\ =\quad \cfrac { 9{ a }^{ 3 }b }{ 3{ a }^{ 2 }b } \\ \\ = \quad 3{ a }$

2. $\left\{ \cfrac { 9{ a }^{ 2 } }{ 3a } \right\} \quad \times \quad \left[ \left( \cfrac { 5a+6a }{ 36 } \right) +{ \left( \cfrac { 4a-a }{ 6 } \right) }^{ 2 } \right]$

$=\quad 3a\quad \times \quad \left[ \cfrac { 11{ a }^{ 2 } }{ 36 } +{ \left( \cfrac { 3a }{ 6 } \right) }^{ 2 } \right] \\ \\ =\quad 3a\quad \times \quad \left[ \cfrac { 11{ a }^{ 2 } }{ 36 } +\cfrac { 9{ a }^{ 2 } }{ 36 } \right] \\ \\ =\quad 3a\quad \times \quad \cfrac { 20{ a }^{ 2 } }{ 36 } \\ \\ =\quad \cfrac { { 60 }{ a }^{ 3 } }{ 36 } \\ \\ =\quad \cfrac { 5{ a }^{ 3 } }{ 3 }$