Algebra · Algebraic manipulation

1 / 8

Simplifying expressions

Why only like terms collect, why a + a and a × a are different animals, and how the index laws drive multiplying, dividing and raising algebraic terms to powers.

Algebra · Algebraic manipulation

Simplifying expressions

Why only like terms collect, why a + a and a × a are different animals, and how the index laws drive multiplying, dividing and raising algebraic terms to powers.

Why it works

A letter stands for a number — so every rule can be tested with numbers. That single idea polices all of algebra. Unsure whether

3a + 2b

"makes"

5ab

? Try

a = 10, b = 1

: left side

32

, right side

50

. Not equal, so the move was illegal.

Only like terms collect.

5a + 3b - 2a + 4b

: the

a

-counts combine and the

b

-counts combine —

3a + 7b

— but

a

's and

b

's can no more merge than a count of apples can absorb a count of bananas. The same guard stops

3a + 2

becoming

5a

: a number term isn't an

a

-term.

Adding counts; multiplying builds powers.

a + a = 2a

(two copies added) while

a \times a = a^2

(copies multiplied). Muddling these is the most common slip in the topic — test with

a = 3

3 + 3 = 6

but

3 \times 3 = 9

.

Multiplying terms: numbers with numbers, letters with letters.

4c \times 3c^2 = 12c^3

— coefficients multiply (

4 \times 3

), and the index law ADDS the powers (

c^1 \times c^2 = c^{1+2}

), because the powers are just tallies of how many

c

's are multiplied together. Same law in reverse for division:

12n^7 \div 3n^2 = 4n^5

— coefficients divide, powers subtract.

A power outside a bracket hits EVERYTHING inside.

(2p^3)^4 = 2^4 \times (p^3)^4 = 16p^{12}

— the 2 gets raised too, and the power-of-a-power law multiplies the indices (

p^3

four times over is

p^{12}

). Leaving the coefficient alone (

2p^{12}

) fails the number test immediately.

Algebraic fractions of terms are just division.

\dfrac{18x^5y^2}{6x^2y} = 3x^3y

: divide the numbers, subtract the powers letter by letter.