Exponents are a shorthand way of telling us that a number is multiplied by itself. The number that is multiplied by itself is called the base. The base is the larger number on the left. The smaller number on the right tells us how many times the number is multiplied by itself. It is called the exponent, or power.
We will usually read it as (base) to the power of (exponent), OR (base) to the (exponent) power.
For example, in the expression 43
4 is the base, while 3 is the exponent. The exponent tells us the number of times the base is to be multiplied by itself. In our example, 4 (the base) is multiplied by itself 3 times (the exponent): 4×4×4 We can call this 4 to the power of 3, or 4 to the third power.
Extra: Since the second and third powers are so common, we have special, short names for them - squared and cubed.
42 can be called simply 4 squared.
43 can be called simply 4 cubed.
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Exponents, or powers, are a shorthand way of writing that a number (the base) is multiplied by itself a certain amount of times (exponent). The exponent itself can be any number. For now, we'll focus on positive, whole numbers.
For example:
42 Is called: 'four to the second power' OR 'four to the power of 2' (or 'four squared'). The 4 will be multiplied by another 4 (two fours). 42=4×4=16
43 Is called: 'four to the third power' OR 'four to the power of 3' (or 'four cubed'). The 4 will be multiplied three times (three fours). 43=4×4×4=64
Remember: The number that is multiplied by itself is called the base, and the number of times the number is multiplied is called the exponent.
Therefore, in the expression 42
4 is the base, while 2 is the exponent. In this case the number 4 is multiplied by itself 2 times, therefore this expression will be called '4 to the second power' OR '4 to the power of 2' OR '4 squared.'
And in the expression 43
4 is the base, while 3 is the exponent. In this case the number 4 is multiplied by itself 3 times, therefore this expression will be called '4 to the third power' OR '4 to the power of 3' or '4 cubed.'
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Question 1
Choose the expression that is equal to the following:
We learned in the Order of Operations that we first solve what's inside the parentheses and then continue with the rest of the equation. What comes next? Let's look at our order of operations acronym: PEMDAS.
The E in PEMDAS is for exponents! They come second and we will solve them after the parentheses and before going on to solve multiplication and division.
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Example exercises using exponents
Exercise 1
Task:
What exponent will make the following equation true?
7□=49
Solution:
We can find the answer using two different approaches.
Method 1 - Replacement:
We can try replacing the unknown exponent with a number, let's say 2, and check if it fits the equation. In our case it seems that we have arrived at the correct answer.
We multiply each of the terms in parentheses by its power.
94×a4×x4+4a×x=
94a4x4+4{ax}
Answer:
94a4x4+4{ax}
Review questions
What is an exponent?
An exponent, or power, is a simple way to say that a number is multiplied by itself. A power has two elements: a base and an exponent. The exponent tells us the number of times the base is going to be multiplied by itself.
Let's see some examples:
Example 1
34=
Here the base is the number 3 and the 4 is the exponent, which means that the number 3 must be multiplied 4 times. Then we have the following:
The five must be multiplied by itself three times, so
53=5×5×5=125
Result
53=125
Why would I use an exponent?
An exponent can help us to simplify the multiplication of the same number. It is a simple way of indicating the number of times that number should be multiplied by itself.
The power of 0: Any number raised to the power of 0 is 1.
A0=1
2. The power of 1: Any number raised to the power of 1 will be the same number.
A1=A
3. Multiplying powers with the same base:
Am×An=Am+n
4. Dividing powers with the same base:
AnAm=Am−n
5. Multiplying powers with the same exponent:
(A×B)n=An×Bn
6. Dividing powers with the same exponent:
(BA)n=BnAn
7. Power of a power:
(Am)n=Am×n
8. Negative power:
A−m=Am1
Using the properties
In order to understand when to use the different properties of exponents, we will need to understand the function of the properties themselves. Let's take a look at some examples:
Example 1
Task:
75×73=
To solve this problem we will use the third property of exponents: multiplying powers with the same base:
75×73=75+3=78
To solve, we have:
78=7×7×7×7×7×7×7×7=5,764,801
Result
75×73=78
Example 2
Task:
8486=
To solve, we will use the fourth property of powers: dividing powers with the same base:
8486=86−4=82
If we want to simplify the power we will get:
82=8×8=64
Result
8486=82
Example 3
Task:
Solve (25)3×2−3=
In the first part we will use the seventh property of powers: power of a power. In the second part we will use the eigth property powers: negative powers.
(25)3=25×3=215
2−3=231
Which gives us:
(25)3×2−3=215×231
Now, we will multiply the fractions
215×231=23215
Finally, we will use the fourth property of powers: dividing powers of the same base:
In the formula, we see that the power shows the number of terms that are multiplied, that is, two times
Since in the exercise we multiply 6 three times, it means that we have 3 terms.
Therefore, the power, which is n in this case, will be 3.
Answer
n=3
Exercise #2
What is the answer to the following?
32−33
Video Solution
Step-by-Step Solution
Remember that according to the order of operations, exponents come before multiplication and division, which come before addition and subtraction (and parentheses always before everything),
So firstcalculate the values of the terms in the power and then subtract between the results:
32−33=9−27=−18Therefore, the correct answer is option A.
Answer
−18
Exercise #3
Sovle:
32+33
Video Solution
Step-by-Step Solution
Remember that according to the order of operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).
So firstcalculate the values of the terms in the power and then subtract between the results:
32+33=9+27=36Therefore, the correct answer is option B.
Answer
36
Exercise #4
Solve the following exercise and circle the correct answer:
52−41=
Video Solution
Step-by-Step Solution
To solve the exercise 52−41=, we need to follow the order of operations, specifically focusing on powers (exponents) before performing subtraction.
Step 1: Calculate 52. This means we multiply 5 by itself: 5×5=25.
Step 2: Calculate 41. Any number to the power of 1 is itself, so 41=4.
Step 3: Subtract the result of 41 from 52: 25−4.
Step 4: Complete the subtraction: 25−4=21.
Thus, the correct answer is 21.
Answer
21
Exercise #5
Solve the following exercise and circle the correct answer:
63−62=
Video Solution
Step-by-Step Solution
To solve the expression 63−62, we will follow the order of operations, which in this case involves evaluating the powers before the subtraction operation.
First, evaluate 63:
63 means 6×6×6.
Calculating this, we get 6×6=36.
Then multiply 36 by 6 to get 36×6=216.
Next, evaluate 62:
62 means 6×6.
Calculating this gives us 36.
Finally, subtract the second result from the first: