# 1.2 Use the language of algebra  (Page 3/18)

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Determine if each is an expression or an equation: $3\left(x-7\right)=27$ $5\left(4y-2\right)-7$ .

equation expression

Determine if each is an expression or an equation: ${y}^{3}÷14$ $4x-6=22$ .

expression equation

Suppose we need to multiply 2 nine times. We could write this as $2·2·2·2·2·2·2·2·2.$ This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write $2·2·2$ as ${2}^{3}$ and $2·2·2·2·2·2·2·2·2$ as ${2}^{9}.$ In expressions such as ${2}^{3},$ the 2 is called the base and the 3 is called the exponent . The exponent tells us how many times we need to multiply the base.

We read ${2}^{3}$ as “two to the third power” or “two cubed.”

We say ${2}^{3}$ is in exponential notation and $2·2·2$ is in expanded notation .

## Exponential notation

${a}^{n}$ means multiply a by itself, n times.

The expression ${a}^{n}$ is read a to the ${n}^{th}$ power.

While we read ${a}^{n}$ as “ a to the ${n}^{th}$ power,” we usually read:

• ${a}^{2}$ a squared”
• ${a}^{3}$ a cubed”

We’ll see later why ${a}^{2}$ and ${a}^{3}$ have special names.

Expression In Words
${7}^{2}$ 7 to the second power or 7 squared
${5}^{3}$ 5 to the third power or 5 cubed
${9}^{4}$ 9 to the fourth power
${12}^{5}$ $12$ to the fifth power

Simplify: ${3}^{4}.$

## Solution

$\begin{array}{cccccc}& & & & & \hfill {3}^{4}\hfill \\ \text{Expand the expression.}\hfill & & & & & \hfill 3·3·3·3\hfill \\ \text{Multiply left to right.}\hfill & & & & & \hfill 9·3·3\hfill \\ \text{Multiply.}\hfill & & & & & \hfill 27·3\hfill \\ \text{Multiply.}\hfill & & & & & \hfill 81\hfill \end{array}$

Simplify: ${5}^{3}$ ${1}^{7}.$

125 1

Simplify: ${7}^{2}$ ${0}^{5}.$

49 0

## Simplify expressions using the order of operations

To simplify an expression    means to do all the math possible. For example, to simplify $4·2+1$ we’d first multiply $4·2$ to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

$\begin{array}{c}\hfill 4·2+1\hfill \\ \hfill 8+1\hfill \\ \hfill 9\hfill \end{array}$

By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.

## Simplify an expression

To simplify an expression , do all operations in the expression.

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. For example, consider the expression:

$4+3·7$

If you simplify this expression, what do you get?

Some students say 49,

$\begin{array}{cccccc}& & & & & \hfill \phantom{\rule{0.8em}{0ex}}4+3·7\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}4+3\phantom{\rule{0.2em}{0ex}}\text{gives}\phantom{\rule{0.2em}{0ex}}7.\hfill & & & & & \hfill \phantom{\rule{0.8em}{0ex}}7·7\hfill \\ \text{And}\phantom{\rule{0.2em}{0ex}}7·7\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}49.\hfill & & & & & \hfill \phantom{\rule{0.8em}{0ex}}49\hfill \end{array}$

Others say 25,

$\begin{array}{cccccc}& & & & & \hfill 4+3·7\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}3·7\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}21.\hfill & & & & & \hfill 4+21\hfill \\ \text{And}\phantom{\rule{0.2em}{0ex}}21+4\phantom{\rule{0.2em}{0ex}}\text{makes}\phantom{\rule{0.2em}{0ex}}25.\hfill & & & & & \hfill 25\hfill \end{array}$

Imagine the confusion in our banking system if every problem had several different correct answers!

The same expression should give the same result. So mathematicians early on established some guidelines that are called the Order of Operations .

## Perform the order of operations.

1. Parentheses and Other Grouping Symbols
• Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
2. Exponents
• Simplify all expressions with exponents.
3. Multiplication and Division
• Perform all multiplication and division in order from left to right. These operations have equal priority.
• Perform all addition and subtraction in order from left to right. These operations have equal priority.
Doing the Manipulative Mathematics activity “Game of 24” give you practice using the order of operations.

4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
divide 3x⁴-4x³-3x-1 by x-3
how to multiply the monomial
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Brandon has a cup of quarters and dimes with a total of 5.55\$. The number of quarters is five less than three times the number of dimes
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
4^×=9
Did anyone else have trouble getting in quiz link for linear inequalities?
operation of trinomial