Simplifying algebra

Algebra involves the use of letters in mathematics. These letters are unknowns and can represent either a single unknown number or a range of unknown numbers.

Sometimes algebraic expressions can be simplified – this means that we collect all the similar terms together. We would never say in conversation “I have 3 apples plus 2 apples”. Instead we would say, “I have 5 apples”. Similarly in algebra we can say:

3\({a}\) + 2\({a}\) = 5\({a}\)

However, if I had 5 bananas and 2 apples, it would be impossible to write this in a simpler way.

In algebra:

5\({b}\) + 2\({a}\) = 5\({b}\) + 2\({a}\)

This cannot be written in a simpler way. When simplifying using addition or subtraction, it is helpful to think of different letters as being completely different things – much like bananas and apples. It is important to note that 5\({b}\) means '5 lots of \({b}\)' or '5 × \({b}\)'.

Here are some more examples of how we can simplify:

7\({b}\) - 4\({b}\) = 3\({b}\)

12\({b}\) + 4 - 3\({b}\) = 4 + 9\({b}\)

2\({z}\) + 3\({y}\) - 7\({z}\) + 6\({y}\) = 9\({y}\) - 5\({z}\)

3\({ab}\) + 2\({a}\) + 7 = 7 + 3\({ab}\) + 2\({a}\)

There are four things to note about the above examples:

the sign (+ or -) belongs to the term that comes after it
when giving our simplified answer we always give it in alphabetical order
a term containing, for example \({ab}\), cannot be added to terms with an \({a}\) or terms with a \({b}\) but must instead be kept separate
numbers on their own cannot be added to terms containing a letter

Question

Simplify 5\({x}\) + 4\({y}\) - 2\({z}\) + 3\({x}\) + \({z}\) - 6\({y}\)

We can also simplify algebraic expressions that involve multiplication. The rules here are very different to the rules for addition and subtraction.

Consider the following:

5\({a}\) × 7\({b}\)

Firstly, we remember that 5\({a}\) = 5 × \({a}\) and 7\({b}\) = 7 × \({b}\)

This leaves us with:

5\({a}\) × 7\({b}\) = 5 × a × 7 × b

This gives the result:

5 × 7 × \({a}\) × \({b}\) = 35\({ab}\)

Sometimes we will have to simplify expressions in the form:

\({a^3}\) × \({a^5}\) or \({d^8}\) × \({d^2}\)

In general \({x^a}\) × \({x^b}\) = \({x^{(a+b)}}\)

This means that when we multiply two terms with indices, the result is that the indices are added.

Examples

\({a^7}\) × \({a^4}\) = \({a}^{7+4}\) = \({a}^{11}\)

\({f^3}\) × \({f^4}\) = \({f^7}\)

\({z^2}\) × \({z^3}\) × \({z^5}\) = \({z}^{10}\)

Or when we have two or more different letters involved:

\({a^3}\) × \({b^4}\) × \({a^2}\) × \({b^7}\) = \({a^3}\) × \({a^2}\) × \({b^4}\) × \({b^7}\) = \({a^5}\) \({b^{11}}\)

\({x^2}\) × \({y^2}\) × \({x^4}\) × \({z^3}\) = \({x^6}\)\({y^2}\)\({z^3}\)

Or when we have a mixture of indices and coefficients:

5\({a^3}\) × 3\({a^2}\) = 5 × 3 × \({a^3}\) × \({a^2}\) = 15\({a^5}\)

Question

Simplify 8\({b}\) × 3\({b}\) × 2\({c}\)

Question

Simplify 6\({b^2}\) × 3\({a^2}\)\({b^3}\)

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Basic algebra – WJECSimplifying algebra