Combining Operations (Distributive Laws)

Summary: Why is it legal to factor 2x+2y as 2(x+y), but not x²+y² as (x+y)²? This page tries to sort out which distributive laws are real and which are not laws at all, but mistakes.

See also: It’s the Law — the Laws of Exponents
It’s the Law Too — the Laws of Logarithms
How to Factor the Sum of Squares

The Problem

Every student learns “the” distributive law at an early age:
 
          3(x + 7)  =  3(x) + 3(7).
 
In fact there are many distributive laws, for instance
 
          (3x)²  =  3² x²
 
Unfortunately, many students also learn all sorts of other distributions that are invalid, like “expanding”
 
          (x − 6)²  =  x² − 6²   
 
Bad habits like that one seem to stick to students like, like, ... well, like unwanted sticky things.

A simple rule can help you remember how you can combine operations and how you cannot—which distributions are legal and which are illegal.

The Solution

Think of a small house. It’s got a basement, a ground floor, and an attic. You can’t jump right from the basement to the attic, can you? But you can take stairs between the basement and ground floor, or between the ground floor and the attic.

You combine operations just like that. If the operations are on adjacent levels, you can combine them; otherwise you can’t. What are the levels? Forget PEMDAS; there are really only three operations to be concerned with:

And the rule is very simple:

You can distribute any operation over an operation one level below it. There are no other distributions.

When you start to distribute one operation over another, stop and ask yourself which distributive law you are using. If it’s not one of the two specific laws mentioned on this page, you’re almost certainly making a mistake.

Examples

Multiply/Divide over Add/Subtract

You can distribute a multiply or divide over an add or subtract, because multiply and divide are one level above add and subtract:
 
          7(x + y)  =  7x + 7y
          (x + y) / 3  =  x/3 + y/3
          2x (x − 3)  =  2x² − 6x
          (2x − 8) / 2  =  2x/2 − 8/2 = x − 4
 
Students sometimes distribute a multiplier over both parts of a fraction, like this:
 
          3 × (2/5)  =  6 / 15   
 
You can’t do that because multiply is not one level above divide; they’re at the same level. You can distribute only when moving down one level.

Sometimes we talk about “distributing a minus sign”, like this:
 
          2x² − (x − 1)  =  2x² − x + 1
 
That is correct because that minus sign for subtracting is the same as adding −1 times the quantity, and what gets distributed is the −1 multiplier:
 
          2x² + (−1)(x − 1)  =  2x² + (−1)x + (−1)(−1)
 
Take a couple of seconds and make sure you see how the first equation is really just a shortcut version of the second.

You probably know that you can not only distribute but collect or “factor out”:
 
          6x + 12  =  6x + 6(2) = 6(x + 2)

Power/Root over Multiply/Divide

You can distribute an exponent or radical over a multiply or divide, because powers and roots are one level above multiply and divide:
 
          (3x)³  =  3³ x³
          √25x  =  √25 √x = 5 √x
          (2/3)²  =  2² / 3² = 4/9
          √x/100  =  √x / √100 = (√x / 10)
 

What you must not do—though students have been doing it since algebra was invented—is to distribute a power or root over an add or subtract:
 
          (x + 3)²  =  x² + 3²   
          √x² − 25  =  x − 5   
 

Look back at the “house” picture. Add/subtract are in the basement, and powers/roots are in the attic. You can’t distribute powers or roots over addition and subtraction because you’d have to skip a level.

Some More No-Nos

You can only distribute down a level, never up:
 
          x^3y  =  x³ x^y   
          2(3x)²  =  (6x)²   

Other Ways to Combine

Yes, you can combine algebra operations in other ways, but the other combinations are never as simple as a distribution. The only straight distributions are the ones mentioned above: distributing an operation one level down in the “house”.

Here’s an example of a combination that is not a straight distribution:
 
          x⁽²⁺³⁾  =  x² + x³   
          x⁽²⁺³⁾  =  x² x³
 
Notice what happens. You can’t distribute addition over a power because addition isn’t one level higher than powers. (It’s not higher at all, but lower, as you know.) But a valid combination does exist: the addition turns into a multiplication.

There are a number of laws for combining power expressions. Ultimately they all trace back to counting, as a separate page explains.

Logarithms and Trig Functions

Remember the “house”? Logarithms and trig functions are not one of those levels. In fact, they’re not in the same building. For example,
 
          sin(A + B)  =  sin A + sin B   
          sin(A + B)  =  sin A cos B + cos A sin B
 
When you try to “distribute” the sine function over a sum, it mutates into something quite strange. And with logarithms, you reach a brick wall:
 
          log(x + y)  =  log x + log y   
          log(x + y) cannot be broken up
 

There are lots of laws for combining trig functions and logarithms with the basic algebraic operations, but none of them is a straight distribution.

For straight distribution, stick to the “house” and its rule of one-level-down, and you’ll be fine.

Test Yourself

Here are a few exercises to test your understanding. Try them yourself with pencil and paper, and then check your answers. Write “true” for the ones that are true. For the false ones (most of them), don’t just write “false” but figure out the correct right-hand side of each equation.

1. √6 = 2 √3
2. x⁵ + 2⁵  =  (x + 2)⁵
3. 6x(x² − 11)  =  6x³ − 66x
4. √(9 − y²)  =  3 − y
5. − (a − b)  =  b − a
6. 3² + 4²  =  5²
7. (3x + 1)²  =  9x² + 1
8. √7 − √5 = √2
9. −5² = 25
10. (a − b)² = a² − b²
11. 
12. 3(x + 1)²  =  (3x+3)²
13. 5 − 2 √5 = 3 √5
14. 3(3x)²  =  9x²
15. √33 = √3 √11
16. 6/5  =  6(1/5)
17. 
18. (a⁻¹ + b⁻¹)⁻¹ = a + b
19. 6² + 5² = 11²
20. 5¹ × 5⁻³ = 25⁻²
21. ³√−16 = −2 ³√2
22. 2x+3 − (2x+2) = 5
23. 
24. x⁵ + x⁴ = x⁹
25. (3x + 2y)⁻² = (3x)⁻² + (2y)⁻²
26. (7x)²  =  49x²
27. −4⁰ = 1
28. √16x⁴  =  4x²
29. 
30. 
31. (1/a + 1/b)^-2 = 1/a² + 1/b²
32. x³ + 7x² = 8x⁵
33. 6¹⁰ ÷ 2¹⁰ = 3¹⁰
34. 3² + 4²  =  7²
35. 4(x²+2x+1)  =  4x²+8x+1

Answers: 3, 5, 6, 15, 16, 21, 26, 28, and 33 are true. The rest are false, and their correct right-hand sides are shown below. (“can’t do” means that there is no simple expansion for the left-hand side.) 1. (√2) (√3)    2. x⁵+32    4. can’t do    7. (3x)²+2(3x)(1)+1² or 9x²+6x+1    8. can’t do    9. −25    10. a²−2ab+b²    11. can’t do    12. 3(x²+2x+1) or 3x²+6x+3    13. can’t do    14. 3(9x²) or 27x²    17. (√10)/9    18. 1/(1/a + 1/b) or ab/(b+a)    19. 36+25 or 61    20. 5⁻² or 1/25    22. 1    23. can’t do    24. x⁴(x+1)    25. 1 / (3x+2y)²    27. −1    29. (x/2)+1    30. 1 / (3x+2y)²    31. 1/(1/a + 1/b)² or a²b²/(b+a)²    32. LHS    34. 9+16 or 25    35. 4x²+8x+4   

Special note on −5²: Excel and some calculators get this wrong, which leads some people to insist that the answer should really be 25, but it’s not. For more, please see Precedence of Unary Operators.

Now here are some examples using log and trig functions. (Just ignore this section if you haven’t studied those subjects.)

1. log 16 − log 8  =  log 2
2. 4 log 3x  =  log 12x
3. 
4. log 11 − log 4 = log 7
5. log(3x/y)  =  log(3x) − log(y)
6. ½ log 64  =  log 32
7. 2 log 15 − log 5 = 2 log 3
8. −log 25 + log 3 = −log 75
9. (½) sin 2D  =  sin D
10. cos(π/2 − A)  =  sin A
11. cos10x − cos 4x  =  cos 6x
12. 
13. 
14. cos(π/2 − A)  =  cos(π/2) − cos A

Answers: 1, 5, 10 are true. The rest are false, and their correct right-hand sides are shown below. (Once again, “can’t do” means that there is no simple expansion for the left-hand side.) 2. log([3x]⁴) or log(81x⁴)    3. log₄ 24; see changing log base    4. can’t do    6. log(64^½) or log 8    7. log(15²/5) or log 45    8. log(3/25)    9. (sin D)(cos D); see double-angle formulas    11. −2(sin 5x)(sin 3x); see sum-to-product formulas    12. can’t do    13. can’t do    14. sin A