# Intermediate Algebra/Expressions and Formulas

## Expressions

Throughout your mathematical journey through Arithmetic, Pre-Algebra, and Geometry, you have been introduced to (and analyzing) equations, and perhaps even expressions, but may have ignored these aspects of math. Equations, of course, involve an equal sign (${\displaystyle =}$) while an expression is merely the calculations involved.

When you simplify an expression, you are answering simple arithmetic problems until you result in the simplest answer, most likely a single number, but sometimes a fraction. When you evaluate an expression, you are utilizing variables to find a single number. Of course, you are familiar with all this from the Arithmetics classes you previously took and the Arithmetic review, correct? If your memory needs some refreshing and some practicing the Order Of Operations (Please Excuse My Dear Aunt Sally), here are some practice problems.

### Practice Problems

Simplify the following expressions.

1. ${\displaystyle 2\!\cdot \!4}$

2. ${\displaystyle 6+3^{2}-14\div \!(3-2^{2})}$

3. ${\displaystyle (-4)^{2}\!\cdot \!3+(42-12)}$

Evaluate the following expressions.

1. ${\displaystyle a+bc^{2}-c(b+ac)^{2}}$
a = 2
b = 3
c = (-3)

2. ${\displaystyle b-(ac)^{2}+(3-b)}$
a = 4
b = (-2)
c = 3

1. ${\displaystyle 2\!\cdot \!4=8}$

2. ${\displaystyle {\begin{matrix}6+3^{2}-14\div \!(-1)&=&6+3^{2}-14\div \!(-1)\\\ &=&6+3^{2}-14\div \!(-1)\\\ &=&6+9-14\div \!(-1)\\\ &=&6+9+14\\\ &=&29\end{matrix}}}$ Don't forget to divide -14 by -1 before adding!

3. ${\displaystyle {\begin{matrix}(-4)^{2}\!\cdot \!3+(42-12)&=&(-4)^{2}\!\cdot \!3+30\\\ &=&16\!\cdot \!3+30\\\ &=&48+30\\\ &=&78\end{matrix}}}$ Don't forget to multiply 16 and 3 before adding!

1. ${\displaystyle {\begin{matrix}a+bc^{2}-c(b+ac)^{2}&=&2+(3)(-3)^{2}-(-3){3+(2)(-3)}^{2}\\\ &=&2+(3)(-3)^{2}-(-3)(3-6)^{2}\\\ &=&2+(3)(-3)^{2}-(-3)(-3)^{2}\\\ &=&2+(3)(9)-(-3)(9)\\\ &=&2+27-(-27)\\\ &=&56\end{matrix}}}$

2. ${\displaystyle {\begin{matrix}b-(ac)^{2}+(3-b)&=&(-2)-{(4)(3)}^{2}+{3-(-2)}\\\ &=&(-2)-(12)^{2}+(3+2)\\\ &=&(-2)-(12)^{2}+5\\\ &=&(-2)-144+5\\\ &=&-141\end{matrix}}}$

## Formulas

Now that we understand how to simplify and evaluate expressions, we can analyze formulas. Formulas are expressions made primarily of variables that are plugged in to evaluate the expression. Formulas are used for science and mathematics often, especially Geometry.

### Practice Problems

Evaluate the following formulas.

1. ${\displaystyle F={\frac {9}{5}}C+32}$ is the formula to convert Fahrenheit to Celsius or vice versa. What would the temperature in Fahrenheit if it was 25 Degrees Celsius?

2. ${\displaystyle C=2\pi \!r}$ is the formula to find the Circumference of a circle with a given radius. Find the Circumference if the diameter was 4 inches long. Remember: 2r = d

3. ${\displaystyle I=PRT\,}$ when I represents interest earned, P represents the principal (starting money), R represents interest rate, and T represents the time, usually express in years. This formula is often used for banking and account. After 2 years, how much interest would have been built up if you placed $5000 into an account that has an interest rate of 9.5% per year? ### Answers 1. ${\displaystyle {\begin{matrix}F&=&\left({\frac {9}{5}}\right)(25)+32\\\ &=&45+32\\\ &=&77\end{matrix}}}$ 2. ${\displaystyle {\begin{matrix}C&=&2\pi \!r\quad AND\quad d&=&2r\\\ &=&4\pi \\\ &\approx &12.56\end{matrix}}}$ Assuming that ${\displaystyle \pi }$ = 3.14 3. ${\displaystyle {\begin{matrix}I&=&(5000)(.095)(2)\\\ &=&950\end{matrix}}}$ Don't forget to change the 9.5% to decimal form! ## Lesson Review Unlike equations, expressions contain no equal sign. In equations, there are two separate expressions that are equal to each other, and you are trying to make both sides of the equal sign... equal. Well, with expressions, you are either simplifying or evaluating them. To simplify an expression, you do as many of the Order of Operations as you can to get to the simplest answer. Meanwhile, to evaluate, you plug in numbers for variables and simplify from there. Formulas are special kinds of expressions and/or equations that are used for science and mathematics. ## Lesson Quiz 1. Evaluate the expression ${\displaystyle ab^{2}\cdot \!(ac)^{2}+c(b-a)}$ if ${\displaystyle a=2}$, ${\displaystyle b=(-3)}$, and ${\displaystyle c=(-2)}$. 2. I put$300 into a savings account that has an Interest Rate of 4.5% per year, and I've had this account for exactly a year now. How much do I have in total (Principal with Interest Earned added on)?

3. To find the area of a trapezoid, we use this formula: ${\displaystyle A={\frac {1}{2}}h(b_{1}+b_{2})}$, where ${\displaystyle h}$ represents the perpendicular height of the trapezoid, ${\displaystyle b_{1}}$ represents one of the bases, and ${\displaystyle b_{2}}$ represents the other base. What would the area of a trapezoid be with ${\displaystyle h=4}$, ${\displaystyle b_{1}=3}$, and ${\displaystyle b_{2}=13}$?

4. Does ${\displaystyle a^{(b^{c})}=(a^{b})^{c}}$?