How to Read Complex Expressions

How to Read Complex Expressions exercise

• 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
  • Addition
  • 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
  • Addition
  • 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".