How to do Order of Operations?

How to do Order of Operations? exercise

• Using the correct order of operations, determine the right recipe.

  Hints

  The expressions $8 - 2 + 5$ and $8 - (2 + 5)$ seem similar. The only difference is the use of parentheses.

  The first expression $8 - 2 + 5$ equals $11$. The second has the result $8 - (2 + 5) = 1$.

  The last step is Addition & Subtraction. Always solve the equation from left to right.

  Solution

  As you know, order is a very important matter in everyday life.

  You might be wondering why we need an order of operations in math, too.

  If the Order of Operations didn't exist, we wouldn't be sure which operation to do first...and this could have a large effect on the answer.

  What kind of operations are covered in the Order of Operations?

  • Parentheses
  • Exponents
  • Multiplication & Division
  • Addition & Subtraction

  As you can see, we have combined the operations Multiplication & Division into one step and Addition & Subtraction into another. If only Multiplication and Division remain to be solved (no Addition or Subtraction), it is very important to perform the operations in the order they appear from left to right.

  By taking the first letter of each of our operations, we get PEMDAS - this way you will never forget the correct Order of Operations again!

• Correctly simplify the expression $(8 - 2)^2 + (5 + 2)$.

  Hints

  When you have simplified parentheses to a single number, you do not have to continue to write the parentheses.

  Remember PEMDAS when evaluating the expressions.

  Parentheses and Exponents come before Addition & Subtraction.

  Solution

  We want to simplify the expression $(8 - 2)^2 + (5 + 2)$. At first, it's helpful to recognize which operations are included in our expression. We can see:

  • two sets of parentheses
  • one exponent
  • addition (twice)
  • and subtraction

  As we know, it's very important to evaluate the expression in the correct order. PEMDAS tells us how to evaluate the expression correctly. We simply have to follow the Order of Operations to arrive at the correct answer:

  1. Parentheses
  2. Exponents
  3. Multiplication & Division
  4. Addition & Subtraction

  We take a look at the parentheses first. We can simplify the first parentheses from $(8 - 2)$ to $6$ and the second parentheses from $(5 + 2)$ to $7$, leaving us with $6^2+7$. As you can see, we have left out the parentheses. We can do that because we do not need them anymore.

  Now, we can simplify the exponent: $6^2 + 7 = 36 + 7$. Finally, we add the remaining numbers, leaving us with $36 + 7 = 43$.

• Correctly simplify the expression and help Timothy figure out Sarah's number.

  Hints

  You can evaluate this expression by following the rules of PEMDAS.

  When evaluating Multiplication & Division or Addition & Subtraction you must evaluate the operations in the order they appear in the equation from left to right.

  Solution

  Sarah had a cool idea. By encoding the last digits of her number, she has given Timothy a really difficult challenge.

  But she didn't know that Timothy would be able to follow the rules of PEMDAS. He was able to simplify the expressions and find out the last three digits!

  This is what he evaluated:

  $\begin{array}{rcl} (2 \times 3)^2+1+800-(14-10)^2 & \overset{P}{=} & 6^2+1+800- 4^2\\ & \overset{E}{=} & 36 +1+800- 16\\ & \overset{A,S}{=} & 821 \end{array}$

  Sarah's complete number is $1-713-555-1821$. Lucky Timothy.

• Find out how many eggs and how much flour Sally needs for her cookie recipe.

  Hints

  Evaluate Parentheses before Exponents.

  When solving a problem, you can often simplify expressions by using PEMDAS.

  Start with the Parentheses. Evaluate the Subtraction operators outside the parentheses in both expressions last.

  You have to follow the rules of PEMDAS inside the parentheses, too.

  Solution

  What we have here is a really tasty recipe for cookies. But someone has substituted the amounts for eggs and flour with some longer mathematical expressions. We have to evaluate the expressions before we can go on baking. Let us take a look and see how many eggs we need:

  $\begin{array}{rcl} (8-\frac{15}{5} \times 2)^3-\frac{(2+3)^2}{25} & \overset{D, A}{\longrightarrow} & (8-3 \times 2)^3-\frac{5^2}{25}\\ & \overset{M}{\longrightarrow} & (8-6)^3-\frac{5^2}{25}\\ & \overset{P}{\longrightarrow} & 2^3- \frac{5^2}{25}\\ & \overset{E}{\longrightarrow} & 8 - \frac{25}{25}\\ & \overset{D}{\longrightarrow} & 8 - 1\\ & \overset{S}{\longrightarrow} & 7 \end{array}$

  We solve the parentheses first. Inside of the left parentheses, we have to simplify the fraction first. Then we multiply $3$ and $2$ and finally subtracted the two numbers. It's very important to follow the rules of PEMDAS inside parentheses, too.

  After evaluating the parentheses, we should move on to the exponents. Now we can simplify the last fraction on the right side before subtracting in the end. That leaves us with $7$ eggs!

  And how much flour do we need? Let's solve the last problem:

  $\begin{array}{rcl} \frac{(6+2)^2}{4^2} \times 3 - \frac{(\frac42) ^3}{2^2} \times 3 & \overset{A,D}{\longrightarrow} & \frac{8^2}{4^2} \times 3 - \frac{2 ^3}{2^2} \times 3\\ & \overset{E}{\longrightarrow} & \frac{64}{16}\times 3 - \frac84 \times 3\\ & \overset{D}{\longrightarrow} & 4 \times 3 - 2 \times 3\\ & \overset{M}{\longrightarrow} & 12 - 6 \\ & \overset{S}{\longrightarrow} & 6 \end{array}$

  For solving this expression, we solved the Parentheses first. Then we can evaluate the Exponents followed by Multiplication and Division. Finally, we subtracted from left to right. We found out that we need $6$ cups of flour.

  Thanks to PEMDAS, we can now start baking!!!

• Identify the mnemonic used to remember PEMDAS.

  Hints

  Look at the initials of the words.

  Which letters are in the acronym PEMDAS?

  Solution

  With PEMDAS you can simplify expressions: you simply have to follow the rules.

  First evaluate Parentheses, then Exponents, followed by Multiplication and Division and finally you evaluate Addition and Subtraction.

  Remember to always evaluate Multiplication and Division as well as Addition and Subtraction from left to right!

  You can remember this order with a funny mnemonic:

  Please Excuse My Dear Aunt Sally.

• Simplify the expressions by using the order of operations.

  Hints

  Follow the rules of PEMDAS.

  PEMDAS stands for:

  • Parentheses
  • Exponents
  • Multiplication & Division
  • Addition & Subtraction

  Sometimes there are Parentheses inside Parentheses. In this case, solve the inner Parentheses first.

  The inside of Parentheses are no different -- follow the rules of PEMDAS.

  Solution

  By following the rules of PEMDAS you cannot fail solving expressions the right way. Let us take a closer look at the following expression. Here we can see Parentheses inside Parentheses.

  $\begin{array}{rcl} 75 - 2 \times \left( 3 + \frac{\left(3+24 \right)}{9} \right)^2 & \overset{P,P,A}{\longrightarrow} & 75 - 2 \times \left( 3 + \frac{27}{9} \right)^2\\ & \overset{P,D}{\longrightarrow} & 75 - 2 \times \left( 3+3 \right)^2\\ & \overset{P,A}{\longrightarrow} & 75 - 2 \times 6^2\\ & \overset{E}{\longrightarrow} & 75 - 2 \times 36\\ & \overset{M}{\longrightarrow} & 75 - 72\\ & \overset{S}{\longrightarrow} & 3 \end{array}$

  We should first look at the expression as a whole. We see that there's another set of Parentheses inside the Parentheses. To solve the outer Parentheses, we must first evaluate the inner Parentheses. Finally, we can solve the outer Parentheses and proceeded as usual, following the rules of PEMDAS.