Exponentials and Whole Numbers - Practice Problems

Background

Some more problems with whole numbers and exponentials.

Question

Evaluate the following:

1.  
2. 
3.  

Answer

1. 
2.  
3.  

Analysis

Question 1

There are a couple of ways we could work this problem: work through all the operations inside the brackets, then take that number to the 0th power, which will result in an answer of 1 (because anything to the 0th power = 1), or skip all that work and know that whatever number it is inside that bracket, the result is going to be 1.

Question 2

There are 3 terms: the fraction with the exponential terms, the multiplication term, and the 6 (which I threw in there just to break things up a bit).

So let's work through a few steps, using order of operations on each of the terms:









Therefore:



Question 3

It's always possible to have operations within the exponent of another term. We treat the operations inside the exponential as a bracket and work it separately from the base:





Therefore:



Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Operations with different kinds of numbers:

• Exponentials and Whole Numbers

Where might we go?

Operations with different kinds of numbers:

• Exponentials and Integers - Practice Problems

Exponentials and Whole Numbers

Background

Let's explore the concept of exponentials some more using whole numbers.

Question

Evaluate:

•  
•  
•  

Answer

• 
•  
•  

Analysis

Question 1



Notice that we have two bases - the 3 and the 5. We can't combine them. So all we can do is evaluate the exponentials and then do the multiplication:





(to work the exponential on a calculator, use the  key):





Therefore



Question 2



With this question, we once again have two different bases, 9 and 3. However, remember that  and so we can rewrite our problem:



We can now combine the bases:







Therefore



Question 3

Let's do that same problem we did in 2, but this time do division and not multiplication:



We already know that this equals:



When we multiply bases together, we add the exponents. And so when we divide bases, we subtract bases:



To show how this works, we can expand out the question using 3s:



Or we can simply evaluate it this way:



Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Numbers:

• Whole Numbers

Operations:

• Exponentials

Where might we go?

Operations with different kinds of numbers:

• Exponentials and Whole Numbers - Practice Problems
• Exponentials and Integers

Multiplication, Division, and Whole Numbers - Practice Problems

Background

Let's work some more problems!

Question

Evaluate:  

 

Answer

Analysis

In these questions, we start by using the Order of Operations:

Question 1

According to the Order of Operations, we do the brackets (the Parentheses) first.



Now we have a multiplication and a division. Let's do that as one step:



And now we can work the addition and subtraction that is left:

36 - 4 + 3 = 32 + 3 = 35

Question 2

Again, we do the brackets first. Note that in the fraction, the numerator has a subtraction in it. The numerator is acting like a set of brackets, so we do the subtraction first before doing the division.



We now have a division (the fraction) and a multiplication. We can do those as one step:



And now we can do the addition:

4 + 10 = 14

Vocabulary used:

• Order of Operations - Mathematical "grammar" that tells us what operations come first. Commonly called PEMDAS and BEDMAS (they both refer to the same set of operations)

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers - Practice Problems
• Multiplication, Division, and Whole Numbers

Where might we go?

Operations with different kinds of numbers:

• Multiplication, Division, and Integers - Practice Problems

Multiplication, Division, and Whole Numbers

Background

We've got Multiplication, Division, and Whole Numbers, so now let's work a problem (there are more in the Practice Problem section).

Question

Find 

Answer

Analysis

Let's first approach this using the Order of Operations. Note that the numerator (top set of terms in the fraction) and the denominator (bottom set of terms in the fraction) act as large brackets, so we don't do the division until we work the operations on the top and bottom.

Let's work the numerator first. Note that there's a bracket (coloured in red), so we work that first:

(4 + 2) x 8

(6) x 8

and now we can work the multiplication:

6 x 8 = 48

Now let's work the denominator. I'll colour the bracket again:

4 x (5 + 7)

4 x (12)

and now we can work the multiplication:

4 x 12 = 48

And now we can work the division:



Vocabulary used:

• Whole Numbers - The set of numbers that start at 0 and increase by 1 (0, 1, 2, 3,...)

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Numbers:

• Whole Numbers

Operations:

• Multiplication
• Division
• Order of Operations (PEDMAS)

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers
• Addition, Subtraction, and Whole Numbers - Practice Problems

Where might we go?

Numbers:

• Fractions and Decimals

Operations with different kinds of numbers:

• Multiplication, Division, and Whole Numbers - Practice Problems
• Multiplication, Division, and Integers