Expressions

6.E.1Exponents01/04

Exponents (whole-number).

Step01of 04

Exponent notation. $b^{n}$ means "multiply $b$ by itself n times." $b$ is the BASE; $n$ is the EXPONENT.

5^{3} = 5 \times 5 \times 5 = 125

6.E.1 — Exponents (whole-number).

Step 01 of 04

Exponent notation. $b^{n}$ means "multiply $b$ by itself n times." $b$ is the BASE; $n$ is the EXPONENT.

5^{3} = 5 \times 5 \times 5 = 125

Step 02 of 04

Squares and cubes.

$x^{2}$	→"x squared" — the base × itself
$x^{3}$	→"x cubed" — the base × itself × itself

"Squared" and "cubed" come from area (square) and volume (cube). A side-3 square has area 9 = 3². A side-3 cube has volume 27 = 3³.

Step 03 of 04

Special cases.

$b^{1}$	→= b (one copy)
$b^{0}$	→= 1 (special definition)
$1^{n}$	→= 1 for any n
$0^{n}$	→= 0 for n > 0

Step 04 of 04

Watch the signs.

$(- 3)^{2}$	→= (−3)(−3) = 9
$- 3^{2}$	→= −(3 × 3) = −9 (exponent binds tighter than the negative)

Key insight

b^{n}

= b multiplied n times. Squares come from area, cubes from volume. Parentheses matter with negatives — exponent binds before the negative sign.

6.E.2 — Write algebraic expressions.

Step 01 of 04

An algebraic expression uses VARIABLES (letters representing unknowns) along with numbers and operations.

Step 02 of 04

Translate words to math.

"5 more than x"	→5 + x or x + 5
"6 less than y"	→y − 6 (NOT 6 − y)
"twice n"	→2n
"k divided by 4"	→k / 4 or k ÷ 4
"3 more than half of x"	→(x/2) + 3

Step 03 of 04

Order matters with subtraction and division. "6 less than y" means start with y, take away 6 → y − 6, NOT 6 − y.

Step 04 of 04

Use parentheses when grouping is implied by the wording.

"4 times the sum of x and 3" → $4 (x + 3)$ , not $4 x + 3$ .

Key insight

Translate words to math directly. Watch for "less than" (reversed order) and grouping ("the sum of"). Variables stand in for unknown numbers.

6.E.3 — Evaluate expressions with variables.

Step 01 of 04

To evaluate an expression, substitute a number for each variable, then compute (using order of operations).

Step 02 of 04

Worked example. Evaluate $3 x + 7$ when $x = 4$ .

3 (4) + 7 = 12 + 7 = 19

Step 03 of 04

Multiple variables. Substitute each one.

5 a - 2 b when a = 6, b = 4 : 5 (6) - 2 (4) = 30 - 8 = 22

Step 04 of 04

Wrap substituted values in parentheses — especially negatives.

x^{2} + 3 x when x = - 2 : (- 2)^{2} + 3 (- 2) = 4 - 6 = - 2

Key insight

Substitute (with parentheses!), then compute using PEMDAS. Wrap negative substitutions in parens to avoid sign mistakes.

6.E.4 — Combine like terms.

Step 01 of 04

Like terms have the SAME variable raised to the SAME power. Combine by ADDING or SUBTRACTING coefficients.

Step 02 of 04

Examples.

3x + 5x	→8x
7y − 2y	→5y
4x + 3y	→CANNOT combine (different variables)
2x² + 5x	→CANNOT combine (different powers)
6 + 5x − 4	→5x + 2

Step 03 of 04

Multi-term examples.

3 x + 5 y - 2 x + 4 y = (3 x - 2 x) + (5 y + 4 y) = x + 9 y

Step 04 of 04

Watch the sign attached to each term. The sign immediately to the LEFT belongs to the term.

8 m - 3 n + 2 m = (8 + 2) m - 3 n = 10 m - 3 n

Key insight

Like terms: same variable, same power. Combine coefficients (with sign). Different variables or powers stay separate.

6.E.5 — Distributive property.

Step 01 of 04

Multiplication distributes over addition and subtraction.

a (b + c) = ab + a c

a (b - c) = ab - a c

Step 02 of 04

Worked examples.

3(x + 4)	→3x + 12
5(2y − 7)	→10y − 35
−2(a + 3)	→−2a − 6
−(x − 5)	→−x + 5 (leading minus = −1)

Step 03 of 04

Distribute, then combine like terms.

2 (3 x + 4) + 5 x = 6 x + 8 + 5 x = 11 x + 8

Step 04 of 04

Reverse direction — factoring. Pull a common factor out front.

6 x + 9 = 3 (2 x + 3)

Key insight

Distribute the OUTSIDE factor to EVERY term inside, including signs. Reverse direction = factor out the GCF. Always combine like terms after distributing.

6.E.6 — Identify equivalent expressions.

Step 01 of 04

Two expressions are equivalent if they give the SAME value for ANY input. You can prove equivalence by simplifying both to the same form, or by testing inputs.

Step 02 of 04

Worked example. Are $3 (x + 2)$ and $3 x + 6$ equivalent?

Distribute: $3 (x + 2) = 3 x + 6$ . Yes, equivalent.

Step 03 of 04

Test method. Pick a few values, compare results.

x = 0	→3(0 + 2) = 6 vs. 3(0) + 6 = 6 ✓
x = 1	→3(1 + 2) = 9 vs. 3(1) + 6 = 9 ✓
x = 5	→3(5 + 2) = 21 vs. 3(5) + 6 = 21 ✓

If they match for several values (especially with different signs), they're almost certainly equivalent. To be sure, simplify algebraically.

Step 04 of 04

Common equivalences.

x + x	→2x
x · x	→x²
2x + 3x	→5x
(x + 2) + (x + 3)	→2x + 5
2(x + 1) + 3	→2x + 5

Key insight

Equivalent = same value for every input. Prove by simplifying both expressions algebraically. Quick test by plugging in a few values, especially x = 0, 1, and a negative.

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