mste final

Product rule for exponent multiplication: $a^m \times a^n$

$$a^{m+n}$$

Power of a power exponent rule: $(a^m)^n$

$$a^{m \times n}$$

Quotient rule for exponent division: $\frac{a^m}{a^n}$

$$a^{m-n}$$

Negative exponent rule: $a^{-n}$

$$\frac{1}{a^n}$$

Fractional exponent rule: $a^{\frac{m}{n}}$

$$\sqrt[n]{a^m} = (\sqrt[n]{a})^m$$

Product rule for logarithms: $\log_b(x \times y)$

$$\log_b(x) + \log_b(y)$$

Quotient rule for logarithms: $\log_b\left(\frac{x}{y}\right)$

$$\log_b(x) - \log_b(y)$$

Power rule for logarithms: $\log_b(x^k)$

$$k \times \log_b(x)$$

Change of base formula for $\log_b(x)$

$$\frac{\log_c(x)}{\log_c(b)} = \frac{\ln(x)}{\ln(b)}$$

$n^{\text{th}}$ term formula of an arithmetic sequence

$$a_n = a_1 + (n - 1) \times d$$

Sum formula for the first $n$ terms of an arithmetic sequence

$$S_n = \frac{n}{2} \times (a_1 + a_n) = \frac{n}{2} \times (2a_1 + (n - 1) \times d)$$

$n^{\text{th}}$ term formula of a geometric sequence

$$a_n = a_1 \times r^{n-1}$$

Finite sum formula of a geometric sequence ($r \neq 1$)

$$S_n = a_1 \times \frac{1 - r^n}{1 - r}$$

Infinite sum formula of a geometric sequence ($|r| < 1$)

$$S_\infty = \frac{a_1}{1 - r}$$

$n^{\text{th}}$ term formula of a harmonic sequence

$$a_n = \frac{1}{a_1 + (n - 1) \times d}$$

Binet's formula for the $n^{\text{th}}$ Fibonacci number ($F_n$)

$F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}$ where $\phi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$

Formula and numerical value of the Golden Ratio ($\phi$)

$$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$$

Algebraic representation for age $t$ years in the past and future

Future age: $A + t$, Past age: $A - t$

Distance formula in motion problems

$$d = r \times t$$

Direct proportionality model for $y$ and $x$

$$y = k \times x$$

Inverse proportionality model for $y$ and $x$

$$y = \frac{k}{x}$$

Joint proportionality model for $y$, $x$, and $z$

$$y = k \times x \times z$$

Combined proportionality model ($y$ directly to $x$, inversely to $z$)

$$y = \frac{k \times x}{z}$$

Proportionality model to the square of $x$

$$y = k \times x^2$$

Value representation of a two-digit number with tens digit $t$ and units digit $u$

$$10 \times t + u$$

Mixture problem balance equation model

$$C_1 \times V_1 + C_2 \times V_2 = C_f \times (V_1 + V_2)$$

Combined work rate formula for time $T$ with individual times $A$ and $B$

$$\frac{1}{T} = \frac{1}{A} + \frac{1}{B}$$

Relative speed between minute and hour hands in clock problems

$$\frac{11}{2}^{\circ} \text{ per minute}$$

Formulas for profit, discount amount, and sale price

$\text{Profit} = \text{Selling Price} - \text{Cost}$, $\text{Discount} = \text{Original Price} \times \text{Rate}$, $\text{Sale Price} = \text{Original Price} - \text{Discount}$

Product rule for exponent multiplication: $a^m \times a^n$

$$a^{m+n}$$

Power of a power exponent rule: $(a^m)^n$

$$a^{m \times n}$$

Quotient rule for exponent division: $\frac{a^m}{a^n}$

$$a^{m-n}$$

Negative exponent rule: $a^{-n}$

$$\frac{1}{a^n}$$

Fractional exponent rule: $a^{\frac{m}{n}}$

$$\sqrt[n]{a^m} = (\sqrt[n]{a})^m$$

Product rule for logarithms: $\log_b(x \times y)$

$$\log_b(x) + \log_b(y)$$

Quotient rule for logarithms: $\log_b\left(\frac{x}{y}\right)$

$$\log_b(x) - \log_b(y)$$

Power rule for logarithms: $\log_b(x^k)$

$$k \times \log_b(x)$$

Change of base formula for $\log_b(x)$

$$\frac{\log_c(x)}{\log_c(b)} = \frac{\ln(x)}{\ln(b)}$$

$n^{\text{th}}$ term formula of an arithmetic sequence

$$a_n = a_1 + (n - 1) \times d$$

Sum formula for the first $n$ terms of an arithmetic sequence

$$S_n = \frac{n}{2} \times (a_1 + a_n) = \frac{n}{2} \times (2a_1 + (n - 1) \times d)$$

$n^{\text{th}}$ term formula of a geometric sequence

$$a_n = a_1 \times r^{n-1}$$

Finite sum formula of a geometric sequence ($r \neq 1$)

$$S_n = a_1 \times \frac{1 - r^n}{1 - r}$$

Infinite sum formula of a geometric sequence ($|r| < 1$)

$$S_\infty = \frac{a_1}{1 - r}$$

$n^{\text{th}}$ term formula of a harmonic sequence

$$a_n = \frac{1}{a_1 + (n - 1) \times d}$$

Binet's formula for the $n^{\text{th}}$ Fibonacci number ($F_n$)

$F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}$ where $\phi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$

Formula and numerical value of the Golden Ratio ($\phi$)

$$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$$

Algebraic representation for age $t$ years in the past and future

Future age: $A + t$, Past age: $A - t$

Distance formula in motion problems

$$d = r \times t$$

Direct proportionality model for $y$ and $x$

$$y = k \times x$$

Inverse proportionality model for $y$ and $x$

$$y = \frac{k}{x}$$

Joint proportionality model for $y$, $x$, and $z$

$$y = k \times x \times z$$

Combined proportionality model ($y$ directly to $x$, inversely to $z$)

$$y = \frac{k \times x}{z}$$

Proportionality model to the square of $x$

$$y = k \times x^2$$

Value representation of a two-digit number with tens digit $t$ and units digit $u$

$$10 \times t + u$$

Mixture problem balance equation model

$$C_1 \times V_1 + C_2 \times V_2 = C_f \times (V_1 + V_2)$$

Combined work rate formula for time $T$ with individual times $A$ and $B$

$$\frac{1}{T} = \frac{1}{A} + \frac{1}{B}$$

Relative speed between minute and hour hands in clock problems

$$\frac{11}{2}^{\circ} \text{ per minute}$$

Formulas for profit, discount amount, and sale price

$\text{Profit} = \text{Selling Price} - \text{Cost}$, $\text{Discount} = \text{Original Price} \times \text{Rate}$, $\text{Sale Price} = \text{Original Price} - \text{Discount}$

Standard form of a linear equation in two variables

$$Ax + By = C$$

Standard form of a linear function

$$f(x) = mx + b$$

Quadratic formula for solving $ax^2 + bx + c = 0$

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Standard form and vertex form of a quadratic function

Standard form: $f(x) = ax^2 + bx + c$ Vertex form: $f(x) = a(x - h)^2 + k$

Intercept form of a straight line equation

$$\frac{x}{a} + \frac{y}{b} = 1$$

Point-slope formula of a straight line equation

$$y - y_1 = m(x - x_1)$$

Slope-intercept form of a straight line equation

$$y = mx + b$$

Slope relationships for parallel and perpendicular lines ($m_1$ and $m_2$)

Parallel lines: $m_1 = m_2$ Perpendicular lines: $m_1 \times m_2 = -1$

Distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Distance formula between parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$

$$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$$

Distance formula from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$

$$d = \frac{|A x_0 + B y_0 + C|}{\sqrt{A^2 + B^2}}$$

Product rule for exponent multiplication: $a^m \times a^n$\n\n

$a^{m+n}$\n\n

Power of a power exponent rule: $(a^m)^n$\n\n

$a^{m \times n}$\n\n

Quotient rule for exponent division: $\frac{a^m}{a^n}$\n\n

$a^{m-n}$\n\n

Negative exponent rule: $a^{-n}$\n\n

$\frac{1}{a^n}$\n\n

Fractional exponent rule: $a^{\frac{m}{n}}$\n\n

$\sqrt[n]{a^m} = (\sqrt[n]{a})^m$\n\n

Product rule for logarithms: $\log\_b(x \times y)$\n\n

$\log\_b(x) + \log\_b(y)$\n\n

Quotient rule for logarithms: $\log\_b\left(\frac{x}{y}\right)$\n\n

$\log\_b(x) - \log\_b(y)$\n\n

Power rule for logarithms: $\log\_b(x^k)$\n\n

$k \times \log\_b(x)$\n\n

Change of base formula for $\log\_b(x)$\n\n

$\frac{\log\_c(x)}{\log\_c(b)} = \frac{\ln(x)}{\ln(b)}$\n\n

$n^{\text{th}}$ term formula of an arithmetic sequence\n\n

$a\_n = a\_1 + (n - 1) \times d$\n\n

Sum formula for the first $n$ terms of an arithmetic sequence\n\n

$S\_n = \frac{n}{2} \times (a\_1 + a\_n) = \frac{n}{2} \times (2a\_1 + (n - 1) \times d)$\n\n

$n^{\text{th}}$ term formula of a geometric sequence\n\n

$a\_n = a\_1 \times r^{n-1}$\n\n

Finite sum formula of a geometric sequence ($r \neq 1$)\n\n

$S\_n = a\_1 \times \frac{1 - r^n}{1 - r}$\n\n

Infinite sum formula of a geometric sequence ($\|r\| < 1$)\n\n

$S\_\infty = \frac{a\_1}{1 - r}$\n\n

$n^{\text{th}}$ term formula of a harmonic sequence\n\n

$a\_n = \frac{1}{a\_1 + (n - 1) \times d}$\n\n

Binet's formula for the $n^{\text{th}}$ Fibonacci number ($F\_n$)\n\n

$F\_n = \frac{\phi^n - \psi^n}{\sqrt{5}}$ where $\phi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$\n\n

Formula and numerical value of the Golden Ratio ($\phi$)\n\n

$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$\n\n

Algebraic representation for age $t$ years in the past and future\n\n

Future age: $A + t$, Past age: $A - t$\n\n

Distance formula in motion problems\n\n

$d = r \times t$\n\n

Direct proportionality model for $y$ and $x$\n\n

$y = k \times x$\n\n

Inverse proportionality model for $y$ and $x$\n\n

$y = \frac{k}{x}$\n\n

Joint proportionality model for $y$, $x$, and $z$\n\n

$y = k \times x \times z$\n\n

Combined proportionality model ($y$ directly to $x$, inversely to $z$)\n\n

$y = \frac{k \times x}{z}$\n\n

Proportionality model to the square of $x$\n\n

$y = k \times x^2$\n\n

Value representation of a two-digit number with tens digit $t$ and units digit $u$\n\n

$10 \times t + u$\n\n

Mixture problem balance equation model\n\n

$C\_1 \times V\_1 + C\_2 \times V\_2 = C\_f \times (V\_1 + V\_2)$\n\n

Combined work rate formula for time $T$ with individual times $A$ and $B$\n\n

$\frac{1}{T} = \frac{1}{A} + \frac{1}{B}$\n\n

Relative speed between minute and hour hands in clock problems\n\n

$\frac{11}{2}^{\circ} \text{ per minute}$\n\n

Formulas for profit, discount amount, and sale price\n\n

$\text{Profit} = \text{Selling Price} - \text{Cost}$, $\text{Discount} = \text{Original Price} \times \text{Rate}$, $\text{Sale Price} = \text{Original Price} - \text{Discount}$\n\n

Standard form of a linear equation in two variables\n\n

$Ax + By = C$\n\n

Standard form of a linear function\n\n

$f(x) = mx + b$\n\n