Mathematics: Levels E, M, D, and A Study Guide for the TABE Test

Expressions and Equations: Part 2

Percentage of Test Level Specifically Assessing Expressions and Equations (— = Assumed)

L	E	M	D	A
not tested	not tested	15%	18%	—

Number Powers

When a number is multiplied by itself, maybe several times, a power of that number is made. The power is shown by using an exponent. Roots and scientific notation are an offshoot of the power concept.

Exponents

As you probably know, an exponent is a number that shows how many times a number or variable is used to multiply by itself. In these examples, $5$ and $x$ are called the bases.

$5^2 = 5 \times 5 = 25$ and $5^3 = 5 \times 5 \times 5 = 125$

$x^2 = x \times x$ and $x^3 = x \times x \times x$

When exponents are used as part of other calculations, you’ll need to know what to do with them. Keep these things in mind:

You can add or subtract numbers with exponents only if they have the same base and the same exponent.
You can add or subtract numbers with exponents only if they have the same base and the same exponent.
- Good: $x^2 + x^2 = 2x^2$
- Bad: $x^2 +x^3 = 2x^5$
  (You can’t add $x^2$ and $x^3$ because they have different exponents.)
- Also bad: $x^2 + y^2 = x^2y^2$
  ($x^2 \text{ and } y^2$ have different bases. You can’t add them.)
When you multiply numbers with the same base, you add their exponents.
- Good: $a^2 \times a^3 = a^5$
- Bad: $a^2 \times a^3 = a^6$
  (Don’t multiply exponents when multiplying numbers.)
If there are coefficients, multiply them. For example, $2x^4\times 3x^5 = 6x^9$.
When you divide numbers with the same base, you subtract their exponents. Thinking of division expressed as a fraction, you subtract the bottom exponent from the top exponent.
- Good: $\frac{c^3}{c^2} = c^{3-2} = c$
- Bad: $\frac{4^2}{4^{-2}} = 4^{2-2} = 4^0=1$
  (It should have been $4^{2-(-2)} = 4^4=256$.)

Roots

Suppose you need to solve $x^2 =36$.

Rewrite it as $x \cdot x = 36$.

This shows that we are looking for a number that, when multiplied by itself, gives $36$. That number is $6$. The number $6$ is the square root of $36$, written $\sqrt{36} = 6$.

What operation needs to be done to each side of the equation $x^2 =36$ to get $x=6$? The answer is taking the square root of both sides, and the steps would look like this:

\[x^2 =36\] \[\sqrt{x^2} =\sqrt{36}\] \[x = 6\]

But wait. We know $6 \times 6 = 36$, but doesn’t $(-6) \times (-6)$ also give $36$? Yes indeed! So the solution to the equation is $x=6$ and $x= -6$. In fact, when we solve equations with $x^2$ in them, we will always end up with two answers, though the two answers can be the same number.

You should know the squares and square roots of small numbers:

\[\begin{array}{cccc} &2^2=4&5^2=25&8^2=64&11^2=121\\ &3^2=9&6^2=36&9^2=81&12^2=144\\ &4^2=16 &7^2 =49&10^2 =100\\ \end{array}\]

What about square roots of numbers that aren’t perfect squares as the ones above are? For example, does $5$ have a square root? The answer is “yes”. It is approximately $2.236$. It’s not possible to write an exact decimal number for $\sqrt{5}$ because there isn’t one. If you square $2.236$, you will get $4.999696$. Close, but not $5$. The same is true of the square roots of all numbers that aren’t perfect squares. These kinds of numbers that can’t be written as exact decimals are called irrational numbers.

In the same way that square roots are needed when solving equations like $x^2 =81$, cube roots are needed to solve equations like $x^3 = 125$.

\[x^3 = 125\] \[x \cdot x \cdot x =125\] \[5 \cdot 5 \cdot 5 = 125\]

We see that $5^3 = 125$, so the cube root of $125$ is $5$. This is written as $\sqrt[3]{125} = 5$.

It will help a lot if you’re familiar with the cubes and cube roots of small numbers:

\[\begin{array}{ccc} &1^3=1&3^3=27&5^3=125\\ &2^3=8&4^3=64\\ \end{array}\]

Scientific Notation

Scientific notation is a way of writing very large and very small numbers. Instead of writing numbers like $3\text{,}000\text{,}000\text{,}000$ or $.000\ 000\ 07$ with all those zeros, powers of ten are used. Here’s how it works.

Start where the decimal point is. If the number is bigger than $10$, move the decimal point to the left until there is only one non-zero digit to its left.

Here is an example:

Start with $3\text{,}000\text{,}000\text{,}000$.
Move the decimal point left until it’s just to the right of $3$ and get $3.000\ 000\ 000$. Drop all the zeros that are now to the right of the decimal point and get $3$.

Count the number of decimal places the decimal point moved over and write that number as a power of $10$. The decimal moved nine places, so we write $10^9$.

Multiply the $3$ and the $10^9$ and get $3 \times 10^9$. That’s it. Scientific notation is always written in just that form: $3 \times 10^9$. The first part is always between $0$ and $10$, and may be a decimal number. For example, the distance from here to the sun is $9.3 \times 10^7$ miles.

But what if the number is very small (less than $1$)? The process is almost the same, but you move the decimal point to the right and make the power of ten negative.

Here is an example:

Start with $.000\ 000\ 07$.

Move the decimal point to the right until it’s just to the right of the $7$ and get $000\ 000\ 07$. Drop all the zeros that are now to the left of the decimal point and get $7$.

Count the number of decimal places the decimal point moved over and write that number as a negative power of $10$. The decimal moved eight places, so we write $10^{-8}$.

Multiply the $7$ and the $10^{-8}$ and get $7 \times 10^{-8}$. That’s it.

You should be able to make comparisons between two different quantities written in scientific notation. Here are a few comparisons to help.

Each exponent of $10$ below is one higher than the one to its left. That means each power of $10$ is ten times the one to its left. Also, each power of $10$ is one hundred times the one that is two places to the left, and one thousand times bigger than the one three places to the left.

\[10^2 < 10^3 < 10^4 <10^5 < 10^6 <10^7<10^8\]

The exponents tell us that $10^8$ is ten times bigger than $10^7$. Likewise, $3 \times 10^8$ is also ten times bigger than $3 \times 10^7$.

But, if we write $6 \times 10^8$, that will be ten times bigger than $3 \times 10^7$ because of the $10^8$, and also two times bigger because of the $6$ being two times the $3$. That makes $6 \times 10^8$ twenty times bigger than $3 \times 10^7$. You multiply both of those factors.

So, to make a comparison, you have to take into account the power of $10$ and the factor of that power.

Mental Math and Estimation

In many cases you will be required to calculate the exact value of an unknown. There are situations, though, where your skill in estimating becomes very useful. You can validate answers by estimating, such as when you have painstakingly computed and arrived at a number, but by estimating you know that the answer you computed doesn’t make sense. This tells you that you have to compute again.

You may also estimate answers when you don’t have the time to do precise calculations. Estimating comes in handy when you need an answer quickly, such as this simple addition:

$153 + 2\text{,}508 + 48 + 3\text{,}091 + 203 + 1\text{,}999 =$ ____ ?

By rounding off the addends, you can do mental addition:

\[150 + 2\text{,}500 + 50 + 3\text{,}100 + 200 + 2\text{,}000 = 8\text{,}000\]

The exact sum is $8\text{,}002$, but $8\text{,}000$ is close enough.

When rounding off, it’s important to round up some numbers and round down some too. This way, you keep the margin of error to a minimum.

A local restaurant goes through $616$ eggs in a week. What would be a good estimate for the number of eggs used in a year?

There are $52$ weeks in a year. If we round that to $50$ and round the $616$ to $600$, we can estimate that the number of eggs in a year would be $50 \times 600 = 30\text{,}000$.

The exact answer is $52 \times 616 = 32\text{,}032$. The fact that the estimate of $30\text{,}000$ and the exact $32\text{,}032$ are fairly close gives us faith that $32\text{,}032$ is very likely the correct answer.

Multi-Step Problems

You should be able to think your way through problems with more than one step, using any whole numbers, fractions, or decimals.

Multi-step word problems can be tricky to do. It helps if you are clear on what you are starting with, and where you have to go. A simple little strategy that can help is to circle the given numbers and underline what is wanted.

Then, read the problem carefully and determine exactly what you’ll be trying to find.

Once you know what you have and where you’re going, the next question is how do you get there? That’s the hard part. There’s no one big strategy that works for every problem, just some tips that can help. If you can’t figure what the first step is, see if there’s anything you can figure out. Any new information you can deduce will help.

Let’s look at an example.

A farmer has $\require{enclose} \enclose{circle}{34}$ chickens, and each one eats $\enclose{circle}{\frac{1}{4}}$ pound of commercial chicken food every day. If he increases his flock to $\enclose{circle}{50}$ chickens, how $\underline{\text{much more chicken food}}$ will the farmer need to buy each week?

How many more chickens does he have now than he did?

If he has $50$ chickens, that’s $50 - 34 = 16$ more than he had.

How much will those $16$ chickens eat in a day?

These $16$ chickens will each eat $\frac{1}{4}$ pound of food a day, for a total of $16 \times \frac{1}{4} = \frac{16}{4} = 4$ pounds of food each day.

In one week that will be $7 \times 4 = 28$ more pounds to buy.

Another strategy that sometimes helps is to start at the end and see if you can work out the step before it. Your thinking can be something like this: “I could find the answer if only I knew the ____.” In the chicken problem you might think “To get the amount of food, I need the number of chickens the farmer added. How can I find that?” That will lead to subtracting the $34$ from $50$.

Here’s another example:

Max saw a video game he wanted for $\$44.95$. He already had $\$21.50$. He was a bagger at a grocery store for $\$8.00$ an hour. To the nearest hour, how many hours will he need to work to buy the video game?

Do the circle and underline strategy.

Max saw a video game he wanted for $\require{enclose} \enclose{circle}{\$44.95}$. He already had $\enclose{circle}{\$21.50}$. He was a bagger at a grocery store for $\enclose{circle}{\$8.00}$ an hour. $\underline{\text{How many hours will he need to work}}$ to buy the video game?

Max would know how many hours to work if he knew how much money he had to make.

He would know how much money he had to make if he subtracted $\$21.50$ from $\$44.95$, so do that first: $\$44.95 -\$21.50 = \$23.45$.

Now, how many $\$8.00$ hours does he need to work to get $\$23.45$? Divide $\$8.00$ into $\$23.45$, and you will get $2.93$. To the nearest hour, we round it to $3$ hours.

Creating Equations

This topic has come up in different ways in this guide, but that’s fine—it’s an important one. It has to do with taking in given information and, using a variable, fitting the information into an equation (or inequality) that can be solved. Let’s take a look at the reasoning in an example:

Claire baked some cookies last week. This week she made $72$ cookies, which was $4$ times as many as last week. How many did she bake last week?

What don’t we know? The number of cookies she baked last week. Call that $x$.

This week she baked $4$ times as many. Write an expression for that: $4x$.

What else do we know about this week? She baked $72$ cookies.

This week she baked $4x$ cookies which we know is $72$. They are equal.

Our equation is $4x=72$.

Solve the equation. Divide both sides by $4$ to get $x=18$.

Here’s another example:

Jordan and Ryan stuffed themselves with pizza. Ryan ate three more slices than Jordan and together they ate $11$ pieces. How many pieces did Jordan eat?

What don’t we know? The number of slices Jordan ate. Call that $x$.
We don’t know how many slices Ryan ate, but it’s $3$ more than Jordan ate. Call it $x+3$.
What about $11$? That’s the total, so we can write $x + x + 3 = 11$.

Solve the equation:

\[x+x+3=11\] \[2x+3=11\] \[2x=8\] \[x=4\]

So, Jordan ate $4$ slices of pizza (and Ryan ate $7$).

Pairs of Equations

Sometimes you will run across equations with two different variables. A single equation like $x+y=12$ shows two numbers adding up to $12$. It can’t be solved except to say that there are an infinite number of solutions that will make it true. For example:

\[(3+9), (4+8), (10+2), (11+1), (6\frac{1}{2} + 5\frac{1}{2}) …\]

But if there are two distinct (not equivalent) equations in $x$ and $y$, we can find a solution:

$x+y=12$
$x-y=6$

The equations show two numbers adding up to $12$, and the same two numbers having a difference of $6$. What two numbers would do both of these things? After thinking for a while, you may come up with $x=9$ and $y=3$.

$9+3=12$
$9-3=6$

This wasn’t super hard to get because the numbers were simple. If the numbers are not so simple, you will need a method for solving the problem. The method involves adding or subtracting the two equations. In our case we can add the two equations and the $y$ variable disappears, leaving an equation we can solve:

\[\begin{array}{llclcr} & &x&+&y&=&12\\ &+&x&-&y&=&6\\ \hline & &2x& & & =&18\\ & &x& & & =&9\\ \end{array}\]

Now that we know $x = 9$, we can substitute $9$ for $x$ in either of the original equations. Let’s choose the first equation:

\[x+y=12\] \[9-y=6\] \[y=9-6\] \[y=3\]

We see again that $x=9$ and $y=3$.

Let’s try another example:

$2x+y=18$
$x+y=11$

Since there are no negative signs, we can’t just add one equation to the other and make a variable disappear. This time we will subtract the second equation from the first:

\[\begin{array}{llclcr} & &2x&+&y&=&18\\ &-&(x&+&y&=&11)\\ \hline & &x& & & =&7\\ \end{array}\]

Now that we know $x = 7$, we can substitute $7$ for $x$ in either of the original equations. Let’s choose the first equation:

\[2x+y=18\] \[14+y=18\] \[y=18-14\] \[y=4\]

Now we know that $x=7$ and $y=4$.

These pairs of equations in two unknowns are called simultaneous equations.

Using Tables and Graphs

We often try to find the relationship between two numbers or variables. Tables and graphs give us visual assistance in doing so. It’s important to know the meaning of the terms used when discussing relationships between numbers or variables.

To review, the independent variable in a relationship is the one being deliberately controlled. The dependent variable is the one that changes as a result of the independent variable change.

For example, suppose a crop scientist is experimenting with fertilizing corn to see how it affects the amount of corn that can be grown in a season. In one garden plot, she puts $5$ lb of fertilizer. In the next three plots she puts $10$ lb, $15$ lb, and $20$ lb. Then, in the fall, she measures the number of bushels of corn in each plot.

There are two things changing here: the weight of fertilizer and the bushels of corn. The scientist was deliberately changing the weight of fertilizer, so that is the independent variable. The amount of corn produced depends on the amount of fertilizer, so that is the dependent variable.

A table can be made of the data from the experiment. It’s customary to put the dependent variable in the first column and the independent variable in the second column. The scientist’s table could look like this:

Fertilizer (lb)	Corn (bu)
5	6
10	12
15	18
20	24

Also, the numbers in the table can be graphed with the independent variable on the horizontal axis. Both ways show that increasing the fertilizer increases the amount of corn grown.

73 Graphing Variables.png

A third common way to show the relationship between two variables is to write an equation. In this case, we can write Yield = Fertilizer weight $\times 1.2$ or, $Y=1.2F$. It might not be obvious that this is the right equation, but if you try putting the numbers from the data table in it, you’ll see that it works. (Quick reality check: It’s incredibly unlikely that any real experiment done with plants and fertilizer would ever have a perfectly linear graph like that. It’s just in math problems that we see that kind of thing.)

Slope

Here is another look at the distance-time graph from earlier in this guide. The fact that the graph is straight and goes through $(0, 0)$ tells us that the relationship is a direct proportion. Let’s calculate the slope of the line to see if it tells us anything interesting.

Remember slope = rise/run.

Picking any two points on the graph, we calculate slope using $\text{slope} = \frac{y_2-y_1}{x_2-x_1}$.

In this case, we’ll use the points $(5, 20)$ and $(10, 40)$.

\[Slope = \frac{40-20}{10-5} = \frac{20}{5} = 4\]

If this was a typical algebra problem with $x, y$ variables, we’d be done, but this graph is showing the motion of an object and we can squeeze a little more information out of it. Both axes represent physical measurements of a real object and they have units.

The $y$-axis has the units in feet, and the $x$ axis has the units in seconds. If we put the units into our calculation, we will have $\frac{ft}{sec}$. That’s a speed. Including the $4$ that we have for a slope, our final conclusion is that the object is moving at a speed of $4\frac{ft}{sec}$.

74 Graphing Slope.png

This kind of interpretation can be done with any graph showing a direct proportion. If we look back at the graph of corn yield and fertilizer, we can calculate its slope to be $1.2$ with units of bushels/pound: $1.2 \frac{\text{bu}}{\text{lb}}$. That tells the grower that for every pound of fertilizer used, there will be $1.2$ bushels of corn grown.

Linear Equations

We’ve seen examples of equations with two variables like $x+y=-7$ or $y=2x-3$, where no exponent of $x$ or $y$ is higher than one. Equations like these always have graphs that are straight lines, so they are called linear equations. Even equations like $y=4$ or $x=6$ have straight line graphs, though they are special examples (the first one is horizontal and the second one is vertical).