Powers

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:

• $4^2$ 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).
  $4^2=4 \times 4=16$
• $4^3$ 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).
  $4^3=4 \times 4 \times 4=64$

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 $4^2$

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 $4^3$

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.'

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.

Take a moment to memorize the acronym PEMDAS for the order of the mathematical operations:

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

$a^n\times a^m=a^{n+m}$

When we multiply exponents with the same base, it's like adding the exponents using simple addition.
For example:

• $4^5=4^{2+3}=4^2 \times 4^3$
• $X^4 \times X^5=X^{4+5}=X^9$

$\left(a\times b\times c\right)^n=a^n\times b^n\times c^n$

For example:

• $(2 \times 3 \times 5)^2 = 2^2 \times 3^2 \times 5^2$
• $(x \times 2 \times x)^2 = x^2 \times 2^2 \times x^2$

• $a^0$
  Any number to the power of $0$ is equal to $1$
• $0^n$
  0 raised to any power (other than zero) is equal to $0$

(Since 0x0=0, 0x0x0=0 and so forth).

• $0^0$
  Zero to the power of zero is undefined.

• $1^n=1$
  One to any power is one.

(Since 1x1=1, 1x1x1=1 and so forth).

• $x^1=x$
  Any number to the power of $1$ stays the same number.

• $6^2=$
• $5^3=$
• $7^3=$
• $({1\over3})^2=$
• $({1\over3})^3=$
• $({2\over3})^3=$
• $({3\over2})^2=$

If you found this article helpful, you may also be interested in the following:

• The Rules of Potentiation
• Division of powers of equal base
• Power of a multiplication
• Power of a quotient
• Power of a power
• Power with zero exponent
• Powers of negative integer exponent
• Taking advantage of all the properties of powers or exponent laws
• Powering integers
• What is a square root and what is it for?

On Tutorela you will find a variety of articles about mathematics.

Task:

What exponent will make the following equation true?

$7^{\square}=49$

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.

$7²=49$

Method 2 - Checking the square root:

$\sqrt{49}=7$

That is

$7²=49$

Answer:

$2$

Task:

$2^{10}\times2^7\times2^6=$

Solution:

When multiplying exponents with the same base, we simply add the powers together.

$10+7+6=23$

Therefore, the solution is:

$2^{23}$

Assignment:

$5^{10}\cdot5^7\cdot5^6=$

As we have learned, when multiplying exponents, if the bases of the exponents are the same you can simply add the exponents:

$10+7+6=23$

$5^{10}\cdot5^7\cdot5^6=5^{23}$

Answer:

$5^{23}$

Task:

Solve the following using like terms:

$6x^6-9x^4=0$

Solution:

First, we must clear the smallest power and simplify the numbers by using the common denominator if possible.

$6x^4\left(x^2-1.5\right)=0$

Separate both sections of the equation that equal to \0\.

We will solve them separately:

$6x^4=0$

Divide by $6x^2$

$x=0$

$x^2-1.5=0$

$x^2=1.5$

$x=\pm\sqrt{\frac{3}{2}}$

Answer:

$x=0,x=\pm\sqrt{\frac{3}{2}}$

Task:

Solve the following equation:

$(A^m)^n$

$(4^X)^2$

Solution:

$(4^X)^2=4^{X\times2}$

Answer:

$4^{x\cdot2}$

Task:

Simplify the following:

$\left(9ax\right)^4+\left(4^a\right)^x$

$(9\times a\times x)^4+(4^a)^x=$

Solution:

We multiply each of the terms in parentheses by its power.

$9^4\times a^4\times x^4+4^{a\times x}=$

$9^4a^4x^4+4^{\left\{ax\right\}}$

Answer:

$9^4a^4x^4+4^{\left\{ax\right\}}$

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:

$3^4=$

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:

$3^4=3\times3\times3\times3=81$

Result

$3^4=81$

$5^3=$

The five must be multiplied by itself three times, so

$5^3=5\times5\times5=125$

Result

$5^3=125$

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.

Let's see a few of the properties of exponents:

1. The power of 0: Any number raised to the power of 0 is $1$.

$A^0=1$

2. The power of 1: Any number raised to the power of 1 will be the same number.

$A^1=A$

3. Multiplying powers with the same base:

$A^m\times A^n=A^{m+n}$

4. Dividing powers with the same base:

$\frac{A^m}{A^n}=A^{m-n}$

5. Multiplying powers with the same exponent:

$\left(A\times B\right)^n=A^n\times B^n$

6. Dividing powers with the same exponent:

$\left(\frac{A}{B}\right)^n=\frac{A^n}{B^n}$

7. Power of a power:

$\left(A^m\right)^n=A^{m\times n}$

8. Negative power:

$A^{-m}=\frac{1}{A^m}$

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:

Task:

$7^5\times7^3=$

To solve this problem we will use the third property of exponents: multiplying powers with the same base:

$7^5\times7^3=7^{5+3}=7^8$

To solve, we have:

$7^8=7\times7\times7\times7\times7\times7\times7\times7=5,764,801$

Result

$7^5\times7^3=7^8$

Task:

$\frac{8^6}{8^4}=$

To solve, we will use the fourth property of powers: dividing powers with the same base:

$\frac{8^6}{8^4}=8^{6-4}=8^2$

If we want to simplify the power we will get:

$8^2=8\times8=64$

Result

$\frac{8^6}{8^4}=8^2$

Task:

Solve $\left(2^5\right)^3\times2^{-3}=$

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.

$\left(2^5\right)^3=2^{5\times3}=2^{15}$

$2^{-3}=\frac{1}{2^3}$

Which gives us:

$\left(2^5\right)^3\times2^{-3}=2^{15}\times\frac{1}{2^3}$

Now, we will multiply the fractions

$2^{15}\times\frac{1}{2^3}=\frac{2^{15}}{2^3}$

Finally, we will use the fourth property of powers: dividing powers of the same base:

$\frac{2^{15}}{2^3}=2^{15-3}=2^{12}$

Result

$\left(2^5\right)^3\times2^{-3}=2^{12}$