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Algebra building blocks — Formula Sheet

6 formulas from Math Foundations, each with what it means, when to use it and the trap it sets.

Exponent laws

x^{a} x^{b} = x^{a + b}, \frac{x ^{a}}{x ^{b}} = x^{a - b}, (x^{a})^{b} = x^{ab}

What it says: Multiplying adds the powers, dividing subtracts them, and a power of a power multiplies them.
When to reach for it: Any expression where the same base appears more than once.
Watch out: These only apply to a shared base. x²·y³ does not simplify, and (x+y)² is not x²+y².

\frac{x ^{7}}{x ^{3}} = x^{4}

Same base, so the exponents subtract.

Negative and fractional exponents

x^{- n} = \frac{1}{x ^{n}}, x^{1/ n} = n x

What it says: A minus sign in the exponent flips the term; a fraction in the exponent is a root.
When to reach for it: Rewriting roots and reciprocals so a power rule can be applied to them.
Watch out: A negative exponent never makes the value negative — it moves the term across the fraction bar.

x = x^{1/2}, \frac{1}{x ^{3}} = x^{- 3}

Both rewrites are what let the power rule reach these later in calculus.

Distributive law

a (b + c) = ab + a c

What it says: A factor outside a bracket multiplies every single term inside it.
When to reach for it: Clearing brackets before collecting like terms.
Watch out: A minus outside flips the sign of every term inside: −(x−3) = −x+3.

- 2 (x - 5) = - 2 x + 10

Both terms take the −2, including the sign.

Solving a linear equation

a x + b = c ⟹ x = \frac{c - b}{a}

What it says: Undo what was done to x, in reverse order, doing the same to both sides.
When to reach for it: One unknown, appearing only to the first power.
Watch out: Whatever you do to one side must happen to the whole other side, not just its first term.

3 x + 4 = 19 \Rightarrow x = \frac{19 - 4}{3} = 5

Subtract, then divide — the reverse of multiply, then add.

Quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

What it says: The two inputs that make ax²+bx+c equal zero.
When to reach for it: A quadratic that will not factor cleanly, or when you need exact roots.
Watch out: The whole numerator sits over 2a, not just the square root. And b² − 4ac negative means there is no real root at all.

x^{2} - 5 x + 6 = 0 \Rightarrow x = \frac{5 \pm 1}{2} = 3, 2

The discriminant is 1, so there are two clean real roots.

Difference of two squares

a^{2} - b^{2} = (a - b) (a + b)

What it says: A subtraction of two perfect squares always splits into these two brackets.
When to reach for it: Anything of the form (something)² − (something)², especially inside a fraction you want to cancel.
Watch out: There is no matching rule for a sum: a² + b² does not factor over the real numbers.

\frac{x ^{2} - 9}{x - 3} = \frac{( x - 3 ) ( x + 3 )}{x - 3} = x + 3

This is the cancellation that rescues most 0/0 limits.

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