Exponential Notation

We use exponential notation to write repeated multiplication. For example [latex]10\cdot10\cdot10[/latex] can be written more succinctly as [latex]10^{3}[/latex]. The 10 in [latex]10^{3}[/latex] is called the base. The 3 in [latex]10^{3}[/latex] is called the exponent. The expression [latex]10^{3}[/latex] is called the exponential expression. Knowing the names for the parts of an exponential expression or term will help you learn how to perform mathematical operations on them.

[latex]\text{base}\rightarrow10^{3\leftarrow\text{exponent}}[/latex]

[latex]10^{3}[/latex] is read as “[latex]10[/latex] to the third power” or “[latex]10[/latex] cubed.” It means [latex]10\cdot10\cdot10[/latex], or [latex]1,000[/latex].

[latex]8^{2}[/latex] is read as “[latex]8[/latex] to the second power” or “[latex]8[/latex] squared.” It means [latex]8\cdot8[/latex], or [latex]64[/latex].

[latex]5^{4}[/latex] is read as “[latex]5[/latex] to the fourth power.” It means [latex]5\cdot5\cdot5\cdot5[/latex], or [latex]625[/latex].

[latex]b^{5}[/latex] is read as “[latex]b[/latex] to the fifth power.” It means [latex]{b}\cdot{b}\cdot{b}\cdot{b}\cdot{b}[/latex]. Its value will depend on the value of [latex]b[/latex].

The exponent applies only to the number that it is next to. Therefore, in the expression [latex]xy^{4}[/latex], only the[latex]y[/latex] is affected by the [latex]4[/latex]. [latex]xy^{4}[/latex] means [latex]{x}\cdot{y}\cdot{y}\cdot{y}\cdot{y}[/latex]. The [latex]x[/latex] in this term is a coefficient of [latex]y[/latex].

If the exponential expression is negative, such as [latex]−3^{4}[/latex], it means [latex]–\left(3\cdot3\cdot3\cdot3\right)[/latex] or [latex]−81[/latex].

If [latex]−3[/latex] is to be the base, it must be written as [latex]\left(−3\right)^{4}[/latex], which means [latex]−3\cdot−3\cdot−3\cdot−3[/latex], or [latex]81[/latex].

Likewise, [latex]\left(−x\right)^{4}=\left(−x\right)\cdot\left(−x\right)\cdot\left(−x\right)\cdot\left(−x\right)=x^{4}[/latex], while [latex]−x^{4}=–\left(x\cdot x\cdot x\cdot x\right)[/latex].

You can see that there is quite a difference, so you have to be very careful! The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication.

The Product Rule for Exponents

What happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[/latex]. Expanding each exponent, this can be rewritten as [latex]\left(2\cdot2\cdot2\right)\left(2\cdot2\cdot2\cdot2\right)[/latex] or [latex]2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2[/latex]. In exponential form, you would write the product as [latex]2^{7}[/latex]. Notice that [latex]7[/latex] is the sum of the original two exponents, [latex]3[/latex] and [latex]4[/latex].

What about [latex]{x}^{2}{x}^{6}[/latex]? This can be written as [latex]\left(x\cdot{x}\right)\left(x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\right)=x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}[/latex] or [latex]x^{8}[/latex]. And, once again, [latex]8[/latex] is the sum of the original two exponents. This concept can be generalized in the following way: For any number [latex]x[/latex] and any integers [latex]a[/latex] and [latex]b[/latex], [latex]\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}[/latex].

The Quotient (Division) Rule for Exponents

Let’s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.

[latex]\displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}[/latex]

You can rewrite the expression as: [latex]\displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}[/latex]. Then you can cancel the common factors of [latex]4[/latex] in the numerator and denominator: [latex]\displaystyle[/latex]

Finally, this expression can be rewritten as [latex]4^{3}[/latex] using exponential notation. Notice that the exponent, [latex]3[/latex], is the difference between the two exponents in the original expression, [latex]5[/latex] and [latex]2[/latex].

So, [latex]\displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[/latex].

Be careful that you subtract the exponent in the denominator from the exponent in the numerator.

So, to divide two exponential terms with the same base, subtract the exponents.

Raise Powers to Powers

Let’s simplify [latex]\left(5^{2}\right)^{4}[/latex]. In this case, the base is [latex]5^2[/latex] and the exponent is [latex]4[/latex], so you multiply [latex]5^{2}[/latex] four times: [latex]\left(5^{2}\right)^{4}=5^{2}\cdot5^{2}\cdot5^{2}\cdot5^{2}=5^{8}[/latex] (using the Product Rule—add the exponents).

[latex]\left(5^{2}\right)^{4}[/latex] is a power of a power. It is the fourth power of [latex]5[/latex] to the second power. And we saw above that the answer is [latex]5^{8}[/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\cdot4=8[/latex].

So, [latex]\left(5^{2}\right)^{4}=5^{2\cdot4}=5^{8}[/latex] (which equals 390,625, if you do the multiplication).

Likewise, [latex]\left(x^{4}\right)^{3}=x^{4\cdot3}=x^{12}[/latex]

This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\left(2^{3}\right)^{5}=2^{15}[/latex].

Raise a Product to a Power

Simplify this expression.

[latex]\left(2a\right)^{4}=\left(2a\right)\left(2a\right)\left(2a\right)\left(2a\right)=\left(2\cdot2\cdot2\cdot2\right)\left(a\cdot{a}\cdot{a}\cdot{a}\cdot{a}\right)=\left(2^{4}\right)\left(a^{4}\right)=16a^{4}[/latex]

Notice that the exponent is applied to each factor of [latex]2a[/latex]. So, we can eliminate the middle steps.

[latex]\begin{array}{l}\left(2a\right)^{4} = \left(2^{4}\right)\left(a^{4}\right)\text{, applying the }4\text{ to each factor, }2\text{ and }a\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,=16a^{4}\end{array}[/latex]

The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.

Raise a Quotient to a Power

Now let’s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex]\displaystyle \frac{3}{4}[/latex] and raise it to the [latex]3[/latex]^rd power.

[latex]\displaystyle {{\left( \frac{3}{4} \right)}^{3}}=\left( \frac{3}{4} \right)\left( \frac{3}{4} \right)\left( \frac{3}{4} \right)=\frac{3\cdot 3\cdot 3}{4\cdot 4\cdot 4}=\frac{{{3}^{3}}}{{{4}^{3}}}[/latex]

You can see that raising the quotient to the power of [latex]3[/latex] can also be written as the numerator ([latex]3[/latex]) to the power of [latex]3[/latex], and the denominator ([latex]4[/latex]) to the power of [latex]3[/latex].

Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.

[latex]\displaystyle {{\left( \frac{a}{b} \right)}^{4}}=\left( \frac{a}{b} \right)\left( \frac{a}{b} \right)\left( \frac{a}{b} \right)\left( \frac{a}{b} \right)=\frac{a\cdot a\cdot a\cdot a}{b\cdot b\cdot b\cdot b}=\frac{{{a}^{4}}}{{{b}^{4}}}[/latex]

When a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below.

[latex]\displaystyle {{\left( \frac{a}{b} \right)}^{4}}=\frac{{{a}^{4}}}{{{b}^{4}}}[/latex]

The Zero Exponent Rule

What if the exponent is zero?

To see how this is defined, let us begin with an example. We will use the idea that dividing any number by itself gives a result of [latex]1[/latex].

[latex]\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1[/latex]

If we were to simplify the original expression using the quotient rule, we would have

[latex]\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[/latex]

If we equate the two answers, the result is [latex]{t}^{0}=1[/latex]. This is true for any nonzero real number, or any variable representing a real number.

[latex]{a}^{0}=1[/latex]

The sole exception is the expression [latex]{0}^{0}[/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined, or [latex]DNE[/latex] (Does Not Exist).

The Negative Exponent Rule

Given a quotient like [latex]\displaystyle \frac{{{2}^{m}}}{{{2}^{n}}}[/latex] what happens when [latex]n[/latex] is larger than [latex]m[/latex]? We will need to use the negative rule of exponents to simplify the expression so that it is easier to understand.

Let’s look at an example to clarify this idea. Given the expression:

[latex]\frac{{h}^{3}}{{h}^{5}}[/latex]

Expand the numerator and denominator, all the terms in the numerator will cancel to [latex]1[/latex], leaving two [latex]h[/latex]s multiplied in the denominator, and a numerator of [latex]1[/latex].

[latex]\begin{array}{l} \frac{{h}^{3}}{{h}^{5}}\,\,\,=\,\,\,\frac{h\cdot{h}\cdot{h}}{h\cdot{h}\cdot{h}\cdot{h}\cdot{h}} \\ \,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}}{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}\cdot {h}\cdot {h}}\\\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{1}{h\cdot{h}}\\\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{1}{{h}^{2}} \end{array}[/latex]

We could have also applied the quotient rule from the last section, to obtain the following result:

[latex]\begin{array}{r}\frac{h^{3}}{h^{5}}\,\,\,=\,\,\,h^{3-5}\\\\=\,\,\,h^{-2}\,\,\end{array}[/latex]

Putting the answers together, we have [latex]{h}^{-2}=\frac{1}{{h}^{2}}[/latex]. This is true when [latex]h[/latex], or any variable, is a real number and is not zero.

Convert Standard Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of [latex]10[/latex]. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on.

Consider the numbers [latex]4000[/latex] and [latex]0.004[/latex]. We know that [latex]4000[/latex] means [latex]4×1000[/latex] and [latex]0.004[/latex] means [latex]4 × \frac{1}{1000}[/latex]. If we write the [latex]1000[/latex] as a power of ten in exponential form, we can rewrite these numbers in this way:

[latex]\begin{array}{cc} \hfill 4000 &&&&&& 0.004 \hfill \\ 4 \text{ x } 1000 &&&&&& 4 \text{ x } \frac{1}{1000} \hfill \\ 4 \text{ x } 10^3 &&&&&& 4 \text{ x } \frac{1}{10^3} \hfill \\ &&&&&& 4 \text{ x } 10^{-3} \hfill \\ \end{array}[/latex]

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than [latex]10[/latex], and the second factor is a power of [latex]10[/latex] written in exponential form, it is said to be in scientific notation.

Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier. If we look at what happened to the decimal point, we can see a method to easily convert from standard notation to scientific notation.

In both cases, the decimal was moved [latex]3[/latex] places to get the first factor, [latex]4[/latex], by itself.

The power of [latex]10[/latex] is positive when the number is larger than [latex]1[/latex]: [latex]4000=4×10^3[/latex].
The power of [latex]10[/latex] is negative when the number is between [latex]0[/latex] and [latex]1[/latex]: [latex]0.004=4×10^{-3}[/latex].

Watch the following video to see more examples of writing numbers in scientific notation.

Convert Scientific Notation to Standard Notation

How can we convert from scientific notation to standard notation? Let’s look at two numbers written in scientific notation and see.

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form – also known as standard notation.

In both cases, the decimal point moved [latex]4[/latex] places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Watch the following video to see more examples of writing scientific notation in standard notation.

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in [latex]1[/latex] L of water. Each water molecule contains [latex]3[/latex] atoms ([latex]2[/latex] hydrogen and [latex]1[/latex] oxygen). The average drop of water contains around [latex]1.32\times {10}^{21}[/latex] molecules of water and [latex]1[/latex] L of water holds about [latex]1.22\times {10}^{4}[/latex] average drops. Therefore, there are approximately [latex]3\cdot \left(1.32\times {10}^{21}\right)\cdot \left(1.22\times {10}^{4}\right)\approx 4.83\times {10}^{25}[/latex] atoms in [latex]1[/latex] L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

Watch the following video to see more examples of multiplying and dividing numbers in scientific notation.