# Simplifying formulas » Basic rules

## Contents

2. Multiplying
3. Multiplying powers
4. Mixed
5. Powers of powers

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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