GED

An exponent indicates repeated multiplication. In 5³, 5 is the base and 3 is the exponent: 5 × 5 × 5 = 125.

ExponentRule	Example
Zeropowerequals1 a⁰=1
Firstpowerequalsthebase	a¹=a
Negativeexponent→reciprocal a⁻ⁿ=1/aⁿ
Multiplysamebases:addexponents	aᵐ×aⁿ=aᵐ⁺ⁿ
Dividesamebases:subtractexponents aᵐ÷aⁿ=aᵐ⁻ⁿ
Powertoapower:multiplyexponents	(aᵐ)ⁿ=aᵐⁿ

A square root is a number that, multiplied by itself, gives the original number: √25 = 5. Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …) have whole-number square roots. Cube roots work similarly: the cube root of 27 is 3 because 3 × 3 × 3 = 27.

Integers include positive whole numbers, negative whole numbers, and zero. On a number line, positive numbers lie to the right of zero and negative numbers to the left. Absolute value is the distance from zero and is never negative: |−7| = 7 and |7| = 7.

Adding integers: same signs, add and keep the sign; different signs, subtract and keep the sign of the number with the larger absolute value.

Subtracting integers: add the opposite of the second number.

Multiplying and dividing integers: same signs give a positive result; different signs give a negative result.

Scientific Notation

Scientific notation writes a number as a value between 1 and 10 multiplied by a power of 10.

Large numbers: move the decimal left and use a positive exponent: 4,500,000 = 4.5 × 10⁶.

Small numbers: move the decimal right and use a negative exponent: 0.00032 = 3.2 × 10⁻⁴.

Operations: to multiply, multiply the coefficients and add the exponents; to divide, divide the coefficients and subtract the exponents.

A point has location but no dimension. A line extends infinitely in both directions; a line segment has two endpoints; a ray has one endpoint and extends infinitely in one direction. An angle is formed by two rays sharing a common endpoint (the vertex) and is measured in degrees.

Complementary angles sum to 90°. Supplementary angles sum to 180°. Vertical angles — the opposite angles formed by two intersecting lines — are always equal.

Parallel lines never intersect and remain the same distance apart. Perpendicular lines intersect at right angles (90°). When a transversal crosses two parallel lines, it creates eight angles with these relationships:

Corresponding angles (same position at each intersection) are equal.

Alternate interior angles (opposite sides of the transversal, between the parallel lines) are equal.

Alternate exterior angles (opposite sides, outside the parallel lines) are equal.

Co-interior (same-side interior) angles are supplementary.

A triangle is a three-sided polygon whose interior angles always sum to 180°.

ClassifiedbySides	Definition
Equilateral	Allsidesequal;allangles60°
Isosceles	Twosidesequal;twoanglesequal
Scalene	Nosidesequal;noanglesequal

ClassifiedbyAngles	Definition
Acute	Allangleslessthan90°
Right	Oneangleequals90°
Obtuse	Oneanglegreaterthan90°

Triangle inequality theorem: the sum of any two sides must be greater than the third side.

Perimeter: the sum of all three sides. Area: A = ½ × base × height, where the height is perpendicular to the base.

Pythagorean Theorem: A² + B² = C²

3² + 4² = 5² c = 5 (hypotenuse) 9 + 16 = 25 a = 3 c is opposite the right angle b = 4

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Figure not to scale.