# Powers - Examples, Exercises and Solutions

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 $4^3$

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): $4\times4\times4$
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.

$4^2$ can be called simply 4 squared.

$4^3$ can be called simply 4 cubed.

Want to learn more? Check out our videos, examples and exercises on this topic!

## Examples with solutions for Powers

### Exercise #1

Sovle:

$3^2+3^3$

### 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 first calculate the values of the terms in the power and then subtract between the results:

$3^2+3^3 =9+27=36$Therefore, the correct answer is option B.

36

### Exercise #2

What is the answer to the following?

$3^2-3^3$

### 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 first calculate the values of the terms in the power and then subtract between the results:

$3^2-3^3 =9-27=-18$Therefore, the correct answer is option A.

$-18$

### Exercise #3

Find the value of n:

$6^n=6\cdot6\cdot6$?

### Step-by-Step Solution

We use the formula: $a\times a=a^2$

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.

$n=3$

### Exercise #4

In the figure in front of you there are 3 squares

Write down the area of the shape in potential notation

### Step-by-Step Solution

Using the formula for the area of a square whose side is b:

$S=b^2$In the picture, we are presented with three squares whose sides from left to right have a length of 6, 3, and 4 respectively:

Therefore the areas are:

$S_1=3^2,\hspace{4pt}S_2=6^2,\hspace{4pt}S_3=4^2$square units respectively,

Consequently the total area of the shape, composed of the three squares, is as follows:

$S_{\text{total}}=S_1+S_2+S_3=3^2+6^2+4^2$square units

To conclude, we recognise through the rules of substitution and addition that the correct answer is answer C.

$6^2+4^2+3^2$

### Exercise #5

$11^2=$

121

### Exercise #6

$6^2=$

36

### Exercise #7

Which of the following is equivalent to the expression below?

$10,000^1$

### Video Solution

$10,000\cdot1$

### Exercise #8

Choose the expression that is equal to the following:

$2^7$

### Video Solution

$2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$

### Exercise #9

What is the missing exponent?

$-7^{\square}=-49$

2

### Exercise #10

$\sqrt{x}=1$

1

### Exercise #11

$\sqrt{x}=2$

4

### Exercise #12

$\sqrt{x}=6$

36

### Exercise #13

$7^3=$

### Video Solution

$343$

### Exercise #14

$5^3=$

### Video Solution

$125$

### Exercise #15

Which of the following clauses is equal to 100?

### Video Solution

$5^2\cdot2^2$