Simplifying formulas » Basic rules

Contents

1. Adding and subtracting
2. Multiplying
3. Multiplying powers
4. Mixed
5. Powers of powers

1. Adding and subtracting

In formula b = 4w + 3w + 6 you can call 4w and 3w like terms.
Like terms can be added together.
In this way the formula can be shortened/simplified to b = 7w + 6.

In formula p = 4a + 2 + 2b are no like terms.
Therefore, this formula cannot be shortened. You write 'cannot be shortened'.
Some teachers may allow you to write 'not possible' or 'n.p.'.
Note: a² and a are not like terms either.

Examples
a + a = 2a
a – a = 0
3a + 7a = 10a
3a + 7b = cannot be shortened
8a – 10a = –2a
20a – a = 19a
8a² + 3a² = 11a²
4a² + 2a = cannot be shortened

2. Multiplying

Let's take 4a × –3a as an example
This multiplication actually means 4 × a × –3 × a
Which can be written as 4 × –3 × a × a = –12a²
Note: Formulas with a ×-sign can always be shortened.

Examples
a × a = a²
3a × a = 3a²
a × b = ab
3a × 4b = 12ab
7a × –2a = –14a²

3. Multiplying powers

When you have to multiply powers, you add the exponents.
a³ × a⁵ = a × a × a × a × a × a × a × a = a^(3 + 5) = a⁸

Examples
7a × a² = 7a³
4a⁴ × 5a² = 20a⁶
a⁴ × b⁷ = a⁴b⁷
3a⁴ × 4a⁵ × –2b⁴ = –24a⁹b⁴

4. Mixed

Of course, you are going to have to reduce formulas where +, –, × and : and powers are used together. Check calculation rules for more information.
Because you have to write intermediate steps for every step in you simplification, it is better to write the steps, just like with equations, underneath each other.

Examples
3x + 4x × 5x =
3x + 20x²

3x² + 4x × 5x =
3x² + 20x² = 23x²

(3x + 4x) × 5x =
7x × 5x = 35x²

4x²(4x – x) – 4x³ =
4x² × 3x – 4x³ =
12x³ – 4x³ = 8x³

5. Powers of powers

If you have a power to a power, this is actually a multiplication.
Remember that a² = a × a. Therefore (ab)² = ab × ab = a²b².

Have a look at the following example:
(3a⁴b²)³ = 3a⁴b² × 3a⁴b² × 3a⁴b² = 27a¹²b⁶
or:
(3a⁴b²)³ = 3³(a⁴)³(b²)³ = 27a¹²b⁶

You probably noticed that the exponents are multiplied.

Examples
(a⁵)³ = a¹⁵
(3a⁴)² = 3²(a⁴)² = 9a⁸

Important:
(–3a³b⁴)⁴ = (–3)⁴(a³)⁴(b⁴)⁴ = 81a¹²b¹⁶

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