Grade 9 Math Review – Roots, Exponents, Geometry & Trigonometry

These flashcards review core Grade 9 math concepts: roots, exponents, modular congruence, irrational numbers, radical operations, geometric theorems, angle relationships, cross multiplication, plane transformations, and basic trigonometric ratios and identities.

What does it mean to be a "root" of a number in mathematics?

It is a value that, when multiplied by itself a specified number of times, produces the original number.

Definition of a square root.

A number that, when multiplied by itself once, equals the original number (e.g., √25 = 5).

Definition of a cube root.

A number that, when multiplied by itself three times, equals the original number (e.g., ∛27 = 3).

Why is √−4 not a real number?

Because the square root of a negative number has no value on the real-number line unless imaginary numbers are used.

How many square roots does any positive number have?

Two: one positive and one negative (e.g., √9 = ±3).

Example: What is √1?

1, because 1 × 1 = 1.

Example: What is ∛64?

4, because 4 × 4 × 4 = 64.

Can roots be irrational? Give an example.

Yes; √2 ≈ 1.41 is irrational.

What is a power in mathematics?

An expression of repeated multiplication using a base and an exponent, written as base^exponent.

Compute 2^3.

8 (2 × 2 × 2).

State the Product Rule for exponents.

a^m · a^n = a^(m+n).

State the Quotient Rule for exponents.

a^m / a^n = a^(m−n), a ≠ 0.

State the Power-of-a-Power Rule.

(a^m)^n = a^(m·n).

What is any non-zero number raised to the 0 power?

1 (a^0 = 1, for a ≠ 0).

Rewrite a^−n using positive exponents.

a^−n = 1 / a^n.

Define congruence in modular arithmetic.

Two numbers are congruent modulo n if they have the same remainder when divided by n.

Interpret: 17 ≡ 5 (mod 12).

17 and 5 both leave a remainder of 5 when divided by 12.

Define an irrational number.

A real number that cannot be expressed as a fraction of two integers; its decimal never ends or repeats.

List two famous irrational numbers.

π (≈3.14159…) and e (≈2.71828…).

Give an example of a rational number mistaken as irrational.

√9 = 3 (actually rational).

Quick test: When is a square root irrational?

If the radicand is not a perfect square and cannot simplify to a fraction.

First step in simplifying √18.

Factor 18 into 9·2 to identify perfect squares.

Simplify √18 completely.

3√2.

Rule for adding radicals.

Only like radicals (same root and radicand) can be combined.

Compute 2√3 + 5√3.

7√3.

Why can’t √2 + √3 be simplified?

They are unlike radicals; radicands differ.

Rule for multiplying radicals √a · √b.

It equals √(a·b).

Multiply √2 · 5.

√10.

State the Midpoint Theorem.

A segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

If D and E are midpoints of AB and AC, what is DE’s relation to BC?

DE ∥ BC and DE = ½ BC.

What are vertically opposite angles?

Equal angles formed opposite each other when two straight lines intersect.

Define a linear pair of angles.

Two adjacent angles that form a straight line and sum to 180°.

Angle sum of any triangle.

180°.

Exterior angle of a triangle equals…

The sum of the two non-adjacent interior angles.

Corresponding angles property for parallel lines cut by a transversal.

They are equal.

Alternate interior angles property.

They are equal.

Co-interior (same-side interior) angles property.

They sum to 180°.

State cross-multiplication for a/ b = c / d.

a·d = b·c.

Explain why cross multiplication works.

Multiplying both sides by the common denominator clears fractions, yielding an equivalent equation.

Important restriction for cross multiplication.

It applies only to equations of two equal fractions, not addition or subtraction of fractions.

Define a plane transformation.

A change in a figure’s position, size, or orientation within the same 2-D plane.

Describe a translation.

Slides every point of a figure the same distance in the same direction; no rotation or reflection.

Describe a rotation.

Turns a figure about a fixed point by a specified angle and direction.

Describe a reflection.

Flips a figure over a line, creating a mirror image.

Describe a dilation.

Rescales a figure larger or smaller from a center point using a scale factor.

Define sine of an angle in a right triangle.

Opposite side ÷ hypotenuse (SOH).

Define cosine of an angle in a right triangle.

Adjacent side ÷ hypotenuse (CAH).

Define tangent of an angle in a right triangle.

Opposite side ÷ adjacent side (TOA).

State the Pythagorean identity for sine and cosine.

sin²θ + cos²θ = 1.

Quotient identity for tangent.

tanθ = sinθ / cosθ.

Reciprocal of sine.

cscθ = 1 / sinθ.

Derived identity: tan²θ + 1 equals…

sec²θ.