NSE Math 10 cheat sheet

Laws of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m-n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^(1/n) = the nth root of a.

Rational exponents: a^(m/n) = (nth root of a)^m = nth root of a^m, valid for a > 0 when n is even.

Scientific notation: write a number as c × 10^k with 1 ≤ |c| < 10; multiply coefficients and add exponents, divide coefficients and subtract exponents.

Rationalizing a denominator: multiply by the conjugate, e.g., 1/(√a + √b) becomes (√a − √b)/(a − b).

Distance on a number line is |a − b|; the midpoint of two real numbers a and b is (a + b)/2.

Irrational numbers such as √2, π, and e have non-terminating, non-repeating decimals; the sum of a rational and an irrational is irrational.

Unit conversion uses ratios equal to 1, e.g., 1 km = 1000 m, so multiply by 1000 m/1 km to cancel units.

For very large or small quantities, compare magnitudes by writing both in the same power of ten before comparing coefficients.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac))/(2a).

Discriminant b² − 4ac: positive gives two real roots, zero gives one repeated real root, negative gives two complex conjugate roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex form is y = a(x − h)² + k with vertex (h, k).

Factoring patterns: a² − b² = (a − b)(a + b); a² + 2ab + b² = (a + b)²; a³ − b³ = (a − b)(a² + ab + b²).

Zero product property: if a product of factors equals zero, at least one factor equals zero, so solve each factor separately.

Solving a system by elimination: multiply equations to match a coefficient, subtract to cancel a variable, then back-substitute.

Exponential growth/decay model: A = P(1 + r)^t for growth and A = P(1 − r)^t for decay, where r is the per-period rate.

When solving radical equations, isolate the radical, square both sides, and check for extraneous roots in the original equation.

A function assigns exactly one output to each input; the vertical line test confirms a graph is a function.

Domain restrictions arise from division by zero (exclude values making the denominator 0) and even roots (require radicand ≥ 0).

Function notation: f(g(x)) means substitute g(x) into f; composition is generally not commutative, so f(g(x)) ≠ g(f(x)).

Inverse functions: f⁻¹ undoes f; to find it, swap x and y and solve for y, and the graphs reflect across the line y = x.

Transformations of y = f(x): f(x) + k shifts up k, f(x − h) shifts right h, a·f(x) stretches vertically by |a|, −f(x) reflects over the x-axis.

Slope of a linear function is (y₂ − y₁)/(x₂ − x₁); parallel lines share slope, perpendicular lines have slopes whose product is −1.

An exponential function f(x) = ab^x has constant ratio between successive outputs; a linear function has constant difference.

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope criteria: lines are parallel when slopes are equal and perpendicular when m₁·m₂ = −1.

Triangle angle sum is 180°; the exterior angle equals the sum of the two remote interior angles.

Similar figures have proportional sides and equal corresponding angles; the ratio of areas equals the square of the scale factor.

Circle facts: circumference C = 2πr, area A = πr², arc length = (θ/360°)·2πr, sector area = (θ/360°)·πr².

Right-triangle relations: Pythagorean theorem a² + b² = c²; special ratios 45-45-90 gives x, x, x√2 and 30-60-90 gives x, x√3, 2x.

Volume formulas: cylinder V = πr²h, cone V = (1/3)πr²h, sphere V = (4/3)πr³, prism V = Bh.

Coordinate geometry: the equation (x − h)² + (y − k)² = r² describes a circle with center (h, k) and radius r.

Mean = sum of values divided by count; median is the middle value of ordered data; mode is the most frequent value.

Interquartile range IQR = Q3 − Q1; outliers are commonly flagged below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures typical distance from the mean; adding a constant to every value leaves the standard deviation unchanged.

Probability of an event = favorable outcomes / total equally likely outcomes, and P(not A) = 1 − P(A).

For independent events, P(A and B) = P(A)·P(B); for mutually exclusive events, P(A or B) = P(A) + P(B).

Conditional probability: P(A|B) = P(A and B)/P(B), the probability of A given that B occurred.

A line of best fit for bivariate data is y = mx + b; the correlation coefficient r ranges from −1 to 1, with sign showing direction.

Random sampling reduces bias; a sample statistic estimates a population parameter, and larger samples generally give more precise estimates.

Modeling cycle: identify the quantity, choose variables, build an equation or function, solve, then interpret and check the result in context.

Linear model y = mx + b: m is the constant rate of change per unit and b is the starting value when the input is zero.

Exponential model y = ab^x: b > 1 models growth, 0 < b < 1 models decay, and b = 1 + r where r is the percent change per period.

Quadratic models describe area and projectile height; maximum or minimum values occur at the vertex, x = −b/(2a).

Rate problems use the relation distance = rate × time; work problems use combined rates as the sum of individual rates.

Check units in every model: the slope of a graph carries units of output per input, and dimensional consistency signals a valid equation.

Proportional reasoning: two quantities are directly proportional when y = kx and inversely proportional when y = k/x.

Interpreting graphs: slope represents a rate, area under a rate graph represents an accumulated total, and intercepts often mark starting or break-even values.

Laws of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m-n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^(1/n) = the nth root of a.

Rational exponents: a^(m/n) = (nth root of a)^m = nth root of a^m, valid for a > 0 when n is even.

Scientific notation: write a number as c × 10^k with 1 ≤ |c| < 10; multiply coefficients and add exponents, divide coefficients and subtract exponents.

Rationalizing a denominator: multiply by the conjugate, e.g., 1/(√a + √b) becomes (√a − √b)/(a − b).

Distance on a number line is |a − b|; the midpoint of two real numbers a and b is (a + b)/2.

Irrational numbers such as √2, π, and e have non-terminating, non-repeating decimals; the sum of a rational and an irrational is irrational.

Unit conversion uses ratios equal to 1, e.g., 1 km = 1000 m, so multiply by 1000 m/1 km to cancel units.

For very large or small quantities, compare magnitudes by writing both in the same power of ten before comparing coefficients.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac))/(2a).

Discriminant b² − 4ac: positive gives two real roots, zero gives one repeated real root, negative gives two complex conjugate roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex form is y = a(x − h)² + k with vertex (h, k).

Factoring patterns: a² − b² = (a − b)(a + b); a² + 2ab + b² = (a + b)²; a³ − b³ = (a − b)(a² + ab + b²).

Zero product property: if a product of factors equals zero, at least one factor equals zero, so solve each factor separately.

Solving a system by elimination: multiply equations to match a coefficient, subtract to cancel a variable, then back-substitute.

Exponential growth/decay model: A = P(1 + r)^t for growth and A = P(1 − r)^t for decay, where r is the per-period rate.

When solving radical equations, isolate the radical, square both sides, and check for extraneous roots in the original equation.

A function assigns exactly one output to each input; the vertical line test confirms a graph is a function.

Domain restrictions arise from division by zero (exclude values making the denominator 0) and even roots (require radicand ≥ 0).

Function notation: f(g(x)) means substitute g(x) into f; composition is generally not commutative, so f(g(x)) ≠ g(f(x)).

Inverse functions: f⁻¹ undoes f; to find it, swap x and y and solve for y, and the graphs reflect across the line y = x.

Transformations of y = f(x): f(x) + k shifts up k, f(x − h) shifts right h, a·f(x) stretches vertically by |a|, −f(x) reflects over the x-axis.

Slope of a linear function is (y₂ − y₁)/(x₂ − x₁); parallel lines share slope, perpendicular lines have slopes whose product is −1.

An exponential function f(x) = ab^x has constant ratio between successive outputs; a linear function has constant difference.

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope criteria: lines are parallel when slopes are equal and perpendicular when m₁·m₂ = −1.

Triangle angle sum is 180°; the exterior angle equals the sum of the two remote interior angles.

Similar figures have proportional sides and equal corresponding angles; the ratio of areas equals the square of the scale factor.

Circle facts: circumference C = 2πr, area A = πr², arc length = (θ/360°)·2πr, sector area = (θ/360°)·πr².

Right-triangle relations: Pythagorean theorem a² + b² = c²; special ratios 45-45-90 gives x, x, x√2 and 30-60-90 gives x, x√3, 2x.

Volume formulas: cylinder V = πr²h, cone V = (1/3)πr²h, sphere V = (4/3)πr³, prism V = Bh.

Coordinate geometry: the equation (x − h)² + (y − k)² = r² describes a circle with center (h, k) and radius r.

Mean = sum of values divided by count; median is the middle value of ordered data; mode is the most frequent value.

Interquartile range IQR = Q3 − Q1; outliers are commonly flagged below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures typical distance from the mean; adding a constant to every value leaves the standard deviation unchanged.

Probability of an event = favorable outcomes / total equally likely outcomes, and P(not A) = 1 − P(A).

For independent events, P(A and B) = P(A)·P(B); for mutually exclusive events, P(A or B) = P(A) + P(B).

Conditional probability: P(A|B) = P(A and B)/P(B), the probability of A given that B occurred.

A line of best fit for bivariate data is y = mx + b; the correlation coefficient r ranges from −1 to 1, with sign showing direction.

Random sampling reduces bias; a sample statistic estimates a population parameter, and larger samples generally give more precise estimates.

Modeling cycle: identify the quantity, choose variables, build an equation or function, solve, then interpret and check the result in context.

Linear model y = mx + b: m is the constant rate of change per unit and b is the starting value when the input is zero.

Exponential model y = ab^x: b > 1 models growth, 0 < b < 1 models decay, and b = 1 + r where r is the percent change per period.

Quadratic models describe area and projectile height; maximum or minimum values occur at the vertex, x = −b/(2a).

Rate problems use the relation distance = rate × time; work problems use combined rates as the sum of individual rates.

Check units in every model: the slope of a graph carries units of output per input, and dimensional consistency signals a valid equation.

Proportional reasoning: two quantities are directly proportional when y = kx and inversely proportional when y = k/x.

Interpreting graphs: slope represents a rate, area under a rate graph represents an accumulated total, and intercepts often mark starting or break-even values.

Laws of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m-n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^(1/n) = the nth root of a.

Rational exponents: a^(m/n) = (nth root of a)^m = nth root of a^m, valid for a > 0 when n is even.

Scientific notation: write a number as c × 10^k with 1 ≤ |c| < 10; multiply coefficients and add exponents, divide coefficients and subtract exponents.

Rationalizing a denominator: multiply by the conjugate, e.g., 1/(√a + √b) becomes (√a − √b)/(a − b).

Distance on a number line is |a − b|; the midpoint of two real numbers a and b is (a + b)/2.

Irrational numbers such as √2, π, and e have non-terminating, non-repeating decimals; the sum of a rational and an irrational is irrational.

Unit conversion uses ratios equal to 1, e.g., 1 km = 1000 m, so multiply by 1000 m/1 km to cancel units.

For very large or small quantities, compare magnitudes by writing both in the same power of ten before comparing coefficients.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac))/(2a).

Discriminant b² − 4ac: positive gives two real roots, zero gives one repeated real root, negative gives two complex conjugate roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex form is y = a(x − h)² + k with vertex (h, k).

Factoring patterns: a² − b² = (a − b)(a + b); a² + 2ab + b² = (a + b)²; a³ − b³ = (a − b)(a² + ab + b²).

Zero product property: if a product of factors equals zero, at least one factor equals zero, so solve each factor separately.

Solving a system by elimination: multiply equations to match a coefficient, subtract to cancel a variable, then back-substitute.

Exponential growth/decay model: A = P(1 + r)^t for growth and A = P(1 − r)^t for decay, where r is the per-period rate.

When solving radical equations, isolate the radical, square both sides, and check for extraneous roots in the original equation.

A function assigns exactly one output to each input; the vertical line test confirms a graph is a function.

Domain restrictions arise from division by zero (exclude values making the denominator 0) and even roots (require radicand ≥ 0).

Function notation: f(g(x)) means substitute g(x) into f; composition is generally not commutative, so f(g(x)) ≠ g(f(x)).

Inverse functions: f⁻¹ undoes f; to find it, swap x and y and solve for y, and the graphs reflect across the line y = x.

Transformations of y = f(x): f(x) + k shifts up k, f(x − h) shifts right h, a·f(x) stretches vertically by |a|, −f(x) reflects over the x-axis.

Slope of a linear function is (y₂ − y₁)/(x₂ − x₁); parallel lines share slope, perpendicular lines have slopes whose product is −1.

An exponential function f(x) = ab^x has constant ratio between successive outputs; a linear function has constant difference.

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope criteria: lines are parallel when slopes are equal and perpendicular when m₁·m₂ = −1.

Triangle angle sum is 180°; the exterior angle equals the sum of the two remote interior angles.

Similar figures have proportional sides and equal corresponding angles; the ratio of areas equals the square of the scale factor.

Circle facts: circumference C = 2πr, area A = πr², arc length = (θ/360°)·2πr, sector area = (θ/360°)·πr².

Right-triangle relations: Pythagorean theorem a² + b² = c²; special ratios 45-45-90 gives x, x, x√2 and 30-60-90 gives x, x√3, 2x.

Volume formulas: cylinder V = πr²h, cone V = (1/3)πr²h, sphere V = (4/3)πr³, prism V = Bh.

Coordinate geometry: the equation (x − h)² + (y − k)² = r² describes a circle with center (h, k) and radius r.

Mean = sum of values divided by count; median is the middle value of ordered data; mode is the most frequent value.

Interquartile range IQR = Q3 − Q1; outliers are commonly flagged below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures typical distance from the mean; adding a constant to every value leaves the standard deviation unchanged.

Probability of an event = favorable outcomes / total equally likely outcomes, and P(not A) = 1 − P(A).

For independent events, P(A and B) = P(A)·P(B); for mutually exclusive events, P(A or B) = P(A) + P(B).

Conditional probability: P(A|B) = P(A and B)/P(B), the probability of A given that B occurred.

A line of best fit for bivariate data is y = mx + b; the correlation coefficient r ranges from −1 to 1, with sign showing direction.

Random sampling reduces bias; a sample statistic estimates a population parameter, and larger samples generally give more precise estimates.

Modeling cycle: identify the quantity, choose variables, build an equation or function, solve, then interpret and check the result in context.

Linear model y = mx + b: m is the constant rate of change per unit and b is the starting value when the input is zero.

Exponential model y = ab^x: b > 1 models growth, 0 < b < 1 models decay, and b = 1 + r where r is the percent change per period.

Quadratic models describe area and projectile height; maximum or minimum values occur at the vertex, x = −b/(2a).

Rate problems use the relation distance = rate × time; work problems use combined rates as the sum of individual rates.

Check units in every model: the slope of a graph carries units of output per input, and dimensional consistency signals a valid equation.

Proportional reasoning: two quantities are directly proportional when y = kx and inversely proportional when y = k/x.

Interpreting graphs: slope represents a rate, area under a rate graph represents an accumulated total, and intercepts often mark starting or break-even values.