# Combining Operations (Distributive Laws)

Copyright © 2002–2020 by Stan Brown

Copyright © 2002–2020 by Stan Brown

**Summary:**
Why is it legal to factor 2*x*+2*y* as 2(*x*+*y*), but not
*x*²+*y*² 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

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

(3*x*)² = 3² *x*²

Unfortunately, many students also learn all
sorts of other distributions that are invalid, like “expanding”

(*x* − 6)² = *x*² − 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.

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:

house floors | operations |
---|---|

attic | powers and roots |

ground floor | multiply and divide |

basement | add and subtract |

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.

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*) = 7*x* + 7*y*

(*x* + *y*) / 3 = *x*/3 + *y*/3

2*x* (*x* − 3) = 2*x*² − 6*x*

(2*x* − 8) / 2 = 2*x*/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:

2*x*² − (*x* − 1) = 2*x*² − *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:

2*x*² + (−1)(*x* − 1) = 2*x*² + (−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”**:

6*x* + 12 = 6*x* + 6(2) = 6(*x* + 2)

You can distribute an exponent or radical over a multiply or
divide, because powers and roots are
**one level above** multiply and divide:

(3*x*)³ = 3³ *x*³

√25*x* = √25 √*x* = 5 √*x*

(2/3)² = 2² / 3² = 4/9

√*x*/100 = √*x* / √100 = (√*x* / 10)

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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 = *x*² + 3²

√*x*² − 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.

You can only **distribute down a level, never up**:

*x*^{3y} = *x*^{3} *x*^{y}

2(3*x*)² = (6*x*)²

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)} = *x*^{2} + *x*^{3}

*x*^{(2+3)} = *x*^{2} *x*^{3}

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.

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.

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.

- √6 = 2 √3
*x*^{5}+ 2^{5}= (*x*+ 2)^{5}- 6
*x*(*x*² − 11) = 6*x*³ − 66*x* - √(9 −
*y*²) = 3 −*y* - − (
*a*−*b*) =*b*−*a* - 3² + 4² = 5²
- (3
*x*+ 1)² = 9*x*² + 1 - √7 − √5 = √2
- −5² = 25
- (
*a*−*b*)² =*a*² −*b*² - 3(
*x*+ 1)² = (3*x*+3)² - 5 − 2 √5 = 3 √5
- 3(3
*x*)² = 9*x*² - √33 = √3 √11
- 6/5 = 6(1/5)
- (
*a*^{−1}+*b*^{−1})^{−1}=*a*+*b* - 6² + 5² = 11²
- 5
^{1}× 5^{−3}= 25^{−2} ^{3}√−16 = −2^{3}√2- 2
*x*+3 − (2*x*+2) = 5 *x*^{5}+*x*^{4}=*x*^{9}- (3
*x*+ 2*y*)^{−2}= (3*x*)^{−2}+ (2*y*)^{−2} - (7
*x*)² = 49*x*² - −4
^{0}= 1 - √16
*x*^{4}= 4*x*^{2} - (1/
*a*+ 1/*b*)^{-2}= 1/*a*² + 1/*b*² *x*^{3}+ 7*x*^{2}= 8*x*^{5}- 6
^{10}÷ 2^{10}= 3^{10} - 3² + 4² = 7²
- 4(
*x*²+2*x*+1) = 4*x*²+8*x*+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. *x*^{5}+32
4. can’t do
7. (3*x*)²+2(3*x*)(1)+1² or 9*x*²+6*x*+1
8. can’t do
9. −25
10. *a*²−2*a**b*+*b*²
11. can’t do
12. 3(*x*²+2*x*+1) or 3*x*²+6*x*+3
13. can’t do
14. 3(9*x*²) or 27*x*²
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. *x*^{4}(*x*+1)
25. 1 / (3*x*+2*y*)²
27. −1
29. (*x*/2)+1
30. 1 / (3*x*+2*y*)²
31. 1/(1/*a* + 1/*b*)² or *a*²*b*²/(*b*+*a*)²
32. LHS
34. 9+16 or 25
35. 4*x*²+8*x*+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
this Dr.
Math article and
this Microsoft
KB article (both accessed 10 June 2010).

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

- log 16 − log 8 = log 2
- 4 log 3
*x*= log 12*x* - log 11 − log 4 = log 7
- log(3
*x*/*y*) = log(3*x*) − log(*y*) - ½ log 64 = log 32
- 2 log 15 − log 5 = 2 log 3
- −log 25 + log 3 = −log 75
- (½) sin 2
*D*= sin*D* - cos(π/2 −
*A*) = sin*A* - cos10
*x*− cos 4*x*= cos 6*x* - 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([3*x*]^{4}) or log(81*x*^{4})
3. log_{4} 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 5*x*)(sin 3*x*); see sum-to-product formulas
12. can’t do
13. can’t do
14. sin *A*

**22 Oct 2020**: Converted to HTML5, replaced obsolete tags with CSS. Improved formatting of radicals, and italicized the variable names.**16 Aug 2015**: Moved from OakRoadSystems.com to BrownMath.com.- (intervening changes suppressed)
- 7 May 2002: New article.

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