# How to Read Complex Expressions

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Susan Sayfan

## DescriptionHow to Read Complex Expressions

More complex expressions often involve multiple operations. For instance, 5(x+2) can be read as “five times the sum of x and two”, and 2/(3-y) can be read as “two divided by the difference of three and y”. With such expressions, it is important to pay attention to the order of operations and to be able to read the expression out loud so that the order of operations is understood. Learn more about how to read complex expressions by helping to protect Megalopolis from the monster llamas. Common Core Reference: CCSS.MATH.CONTENT.6.EE.A.2

### TranscriptHow to Read Complex Expressions

Megalopolis is on lockdown! Why? Every ten years, monster llamas, looking to graze, chow through the fields and forests surrounding Megalopolis. But this year, they're headed straight for the city! To fend off the llama monsters, the mayor of Megalopolis launches Neo-Hunter, a mecha for dealing with pretty much anything. But a new pilot, Aerial Earmath, has just completed her training and she’s a little unsure of how to work the guidance system. Luckily for Aerial, her robot friend, K2B2, will help her figure out how to read complex expressions so she can initiate the defense system.

### First example - Addition and multiplication

Aerial sees an algebraic expression on the small monitor. To input the coordinates of the launch, she reads the expression: 3 times n plus 6. Oh no! That's wrong! An error message?! Her robot pal, K2B2, is trying to tell her something. In math, certain keywords - such as sum, difference, product and quotient tell you how the expressions are grouped. The correct way to read any algebraic expression is to distinguish what operation should be done and in what order. Thanks to K2B2, we know to group the addition before applying the multiplication and we're able to read the expression correctly, ‘3 times the sum of n and 6.’ Great work! The Neo-Hunter is now programmed to lock onto the correct coordinates but with no time to rest on her laurels Aerial gets another set of coordinates.

### Second example - Subtraction and division

This time the expression includes division. At first, Aerial reads out the coordinates as, ’three minus n divided by 5.’ Oh no! Aerial didn't group the operations for this algebraic expression correctly. For fractions, we should read the numerator first, and then read the denominator. To help you stay organized when reading expressions, you can imagine that the numerator has parentheses around it, but it's not entirely necessary. For this example, the coordinates should be read: 'the quotient of the difference of 3 and n and 5.' Aerial thinks one more strike will get rid of the llamas for good.

### Third example - Expression in written words

The next coordinate is an expression written out in words! We lucked out! All we have to do is read the expression and see what coordinates come out. ‘The quantity, x plus 8, raised to the 4th power.’ Let's start with what we know: 'x' plus 8. The word 'quantity' lets us know that this portion of the expression should be evaluated before any other operation is performed. In this case raising it to the 4th power. It worked! Neo-Hunter launches the projectiles of pizza? The city is saved! Hip, hip hooray! The llama monsters happily chow down.

## How to Read Complex Expressions Exercise

Would you like to practice what you’ve just learned? Practice problems for this video How to Read Complex Expressions help you practice and recap your knowledge.
• ### Explain how to read algebraic expressions out loud.

Hints

Keep the parentheses in mind: there is a difference between $3(n+6)$ and $3n+6$.

In a fraction, you can also put the numerator or the denominator in parentheses:

$\frac{3-n}5=\frac{(3-n)}{5}$.

You first have to consider the term inside the parentheses.

For the order of operations remember:

• Paranthesis
• Exponent
• Multiplication
• Division
• Subtraction
Solution

Each time we have to decide which operations should be done and the order in which they are done.

First Aerial tries the following:

"Three times $n$ plus six!"

But, what's that? The login failed! Why?

The statement above stands for $3n+6$ without the parentheses. But there are parentheses around the sum of $n$ and $6$. So you have to read as follows:

"Three times the sum of $n$ and six."

In a similar way, if we read the second term as "Three minus $n$ over five“ then what we actually said was $3-\frac n5$.

So you have to show clearly that we are dividing a difference: "The quotient of the difference of 3 and $n$ and five."

• ### Represent the given expression in an algebraic form.

Hints

"... raise to the $4^{th}$ power" means $(~~~)^4$.

"plus" indicates addition ($+$).

There is a difference between $x^4+8^4$ and $(x+8)^4$.

You should read $x^4+8^4$ as follows:

"$x$ raised to the $4^{th}$ power plus $8$ raised to the $4^{th}$ power."

Solution

"The quantity $x$ plus $8$, raised to the $4^{th}$ power."

What does this mean? Let's look at the expression step by step:

• "The quantity $x$ plus $8$" indicates addition. So we have $x+8$.
• "... raised to the $4^{th}$ power" means $(~~~)^4$.
But what should we raise to the $4^{th}$ power: $8$ or $x+8$? It would be $x+8$. Otherwise, it would read "$x$ plus $8$ to the $4^{th}$ power".

Thus, we get the following algebraic expression: $(x+8)^4$.

• ### Identify each algebraic expression with the correct way of reading it out loud.

Hints

Pay attention to the paranthesis.

Keep PEMDAS in mind for the order of operations.

• Paranthesis
• Exponent
• Multiplication
• Division
• Subtraction

Here you see

"Three times the difference of $n$ and two."

Solution

For transforming expressions in algebraic terms we have to identify the operations as well as the order of those operations.

• $n-4\cdot 3$; the priority rule is Multiplication before Difference: "$n$ minus four times three."
• $5\cdot (n+4)$; the priority rule is Paranthesis before Multiplication: "Five times the sum of $n$ and four."
• $\frac{3+n}4$; the priority rule is Paranthesis before Division: "The quotient of the sum of three and $n$ and four".
• $\frac 3 4+n$; the priority rule is Divsion before Addition: "The quotient of three and four plus $n$."
• ### Determine the right expression in written form for each of the given audio clips.

Hints

plus or sum indicate addition ($+$).

quotient or over stand for division ($\div$).

times indicates multiplication ($\times$).

• $3(n+4)$ can be read as "three times the sum of $n$ and four" or "three times the quantity of $n$ plus four"
• $3n+4$ can be read as "three times $n$ plus four"
Solution

Let's start with "Three times the sum of two times $x$ and three". We can see that "...the sum of two times $x$ and three" can be written as $2x+3$. Then we write three times this sum. To indicate the order we use parentheses around the sum: $3(2x+3)$.

Next we examine "the quotient of the sum of three and $x$ and the sum of two and $x$". "The quotient o...f" indicates $(~~~)\div(~~~)$. The numerator as well as the denominator are sums: $3+x$ and $2+x$. Together we get $\frac{3+x}{2+x}=(3+x)\div(2+x)$.

The last expression is "Three times $x$ plus two times the sum of $x$ and two". We have that "Three times $x$..." can be written as $3x$, "...the sum of $x$ and two" is $x+2$, and "...two times the sum of $x$ and two" is the algebraic term $2(x+2)$. Putting this all together gives us $3x+2(x+2)$.

• ### Name keywords that indicate operations.

Hints

You can use numbers in algebraic operations: $5+6$.

You can also multiply numbers by variables: $5\times x$.

Keep the following terms in mind:

• summand $+$ summand $=$ sum
• minuend $-$ subtrahend $=$ difference
• factor $\times$ factor $=$ product
• dividend $\div$ divisor $=$ quotient
Solution

To transform a given expression in an algebraic term, we first have to decide which operations are used.

There are a lot of keywords indicating the use of an operation:

• sum or plus indicate addition: $+$
• difference indicates subtraction: $-$
• times indicates multiplication: $\times$
• quotient indicates division: $\div$
• ### Find the right audio clip(s) for the expression.

Hints

You can read a fraction as "the quotient of ... and ...", or as "the quotient of ... over ....".

• Three times the sum of $x$ and two can be written as $3(x+2)$.
• Three times $x$ and two can be written as $3x+2$.

Keep the order of operations as well as paranthesis in mind.

Solution

Let's start with the left part: it's a fraction with a sum in the numerator and a difference in the denominator:

• $3x+4$ can be read as "three $x$ plus four", or "the sum of three $x$ and four"
• $x-5$ is "the difference of $x$ and five", or "$x$ minus five"
So we can read the fraction as "the sum of $3x$ and four over the difference of $x$ and five". We can also read is as "the quotient of the sum of three $x$ and four and the difference of $x$ and five".

Next we consider the right term, $-5(2x-3)$, which can be read as "minus five times the difference of two $x$ and 3".

Now we can put those parts together to get,

"The quotient of the sum of three $x$ and four and the difference of $x$ and five minus five times the difference of two $x$ and three",

or

"The sum of three $x$ and four over the difference of $x$ and five minus five times the difference of two $x$ and three".