Properties of Exponents and Scientific Notation

Simplify expressions using the properties for exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression $a^m$, the exponent $m$ tells us how many times we use the base $a$ as a factor.

Let’s review the vocabulary for expressions with exponents.

When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

First, we will look at an example that leads to the Product Property. Consider $x^2 \cdot x^3$. What does this mean?

Notice that $5$ is the sum of the exponents, $2$ and $3$. We see $x^2 \cdot x^3$ is $x^{2+3}$, or $x^5$. The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Example. Simplify each expression: (a) $y^5 \cdot y^6$ (b) $2^x \cdot 2^{3x}$ (c) $2a^7 \cdot 3a$ (d) $d^4 \cdot d^5 \cdot d^2$.

(a)

(b)

(c)

(d)

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples. Consider $\tfrac{x^5}{x^2}$ and $\tfrac{x^2}{x^3}$.

Notice, in each case the bases were the same and we subtracted the exponents. We see $\tfrac{x^5}{x^2}$ is $x^{5-2}$ or $x^3$. When the larger exponent was in the numerator, we were left with factors in the numerator. We see $\tfrac{x^2}{x^3}$ is $\tfrac{1}{x}$; when the larger exponent was in the denominator, we were left with factors in the denominator. This leads to the Quotient Property for Exponents.

Example. Simplify each expression: (a) $\tfrac{x^9}{x^7}$ (b) $\tfrac{3^{10}}{3^2}$ (c) $\tfrac{b^8}{b^{12}}$ (d) $\tfrac{7^3}{7^5}$.

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

(a) Since $9 > 7$, there are more factors of $x$ in the numerator.

(b) Since $10 > 2$, there are more factors of $3$ in the numerator.

(c) Since $12 > 8$, there are more factors of $b$ in the denominator.

(d) Since $5 > 3$, there are more factors of $7$ in the denominator.

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like $\tfrac{a^m}{a^m}$. We know $\tfrac{x}{x} = 1$, for any $x$ ($x \neq 0$) since any number divided by itself is $1$. The Quotient Property tells us that $\tfrac{a^m}{a^m}$ simplifies to $a^0$ and to $1$. So $a^0 = 1$. Any non-zero base raised to the power of zero equals $1$.

In this text, we assume any variable that we raise to the zero power is not zero.

Example. Simplify each expression: (a) $9^0$ (b) $n^0$.

The definition says any non-zero number raised to the zero power is $1$.

Use the definition of a negative exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. What if we just subtract exponents regardless of which is larger?

Let’s consider $\tfrac{x^2}{x^5}$. We subtract the exponent in the denominator from the exponent in the numerator. We see $\tfrac{x^2}{x^5}$ is $x^{2-5}$ or $x^{-3}$. We can also simplify $\tfrac{x^2}{x^5}$ by dividing out common factors:

This implies that $x^{-3} = \tfrac{1}{x^3}$ and it leads to the definition of a negative exponent. If $n$ is an integer and $a \neq 0$, then $a^{-n} = \tfrac{1}{a^n}$.

Let’s now look at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

This implies $\tfrac{1}{a^{-n}} = a^n$ and is another form of the definition of Properties of Negative Exponents.

The negative exponent tells us we can rewrite the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.

Example. Simplify each expression: (a) $x^{-5}$ (b) $10^{-3}$ (c) $\tfrac{1}{y^{-4}}$ (d) $\tfrac{1}{3^{-2}}$.

(a)

(b)

(c)

(d)

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

This tells us that $\left(\tfrac{3}{4}\right)^{-2} = \left(\tfrac{4}{3}\right)^2$. To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base — the fraction — and changed the sign of the exponent. This leads us to the Quotient to a Negative Power Property.

Example. Simplify each expression: (a) $\left(\tfrac{5}{7}\right)^{-2}$ (b) $\left(-\tfrac{x}{y}\right)^{-3}$.

(a)

(b)

Now that we have negative exponents, we will use the Product Property with expressions that have negative exponents.

Example. Simplify each expression: (a) $z^{-5} \cdot z^{-3}$ (b) $(m^4 n^{-3})(m^{-5} n^{-2})$ (c) $(2x^{-6}y^8)(-5x^5y^{-3})$.

(a)

(b)

(c)

Now let’s look at an exponential expression that contains a power raised to a power. Consider $(x^2)^3$.

Notice the $6$ is the product of the exponents, $2$ and $3$. We see that $(x^2)^3$ is $x^{2 \cdot 3}$ or $x^6$. We multiplied the exponents. This leads to the Power Property for Exponents.

Example. Simplify each expression: (a) $(y^5)^9$ (b) $(4^4)^7$ (c) $(y^3)^6 (y^5)^4$.

(a)

(b)

(c)

We will now look at an expression containing a product that is raised to a power. Consider $(2x)^3$.

Notice that each factor was raised to the power and $(2x)^3$ is $2^3 \cdot x^3$. The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

Example. Simplify each expression: (a) $(-3mn)^3$ (b) $(-4a^2b)^0$ (c) $(6k^3)^{-2}$ (d) $(5x^{-3})^2$.

(a)

(b)

(c)

(d)

Now we will look at an example that will lead us to the Quotient to a Power Property. Consider $\left(\tfrac{x}{y}\right)^3$.

Notice that the exponent applies to both the numerator and the denominator. We see that $\left(\tfrac{x}{y}\right)^3$ is $\tfrac{x^3}{y^3}$. This leads to the Quotient to a Power Property for Exponents.

Example. Simplify each expression: (a) $\left(\tfrac{b}{3}\right)^4$ (b) $\left(\tfrac{k}{j}\right)^{-3}$ (c) $\left(\tfrac{2xy^2}{z}\right)^3$ (d) $\left(\tfrac{4p^{-3}}{q^2}\right)^2$.

(a)

(b)

(c)

(d)

We now have several properties for exponents. Let’s summarize them.

Let’s do some more examples that use more than one of the properties.

Example. Simplify each expression by applying several properties: (a) $(3x^2 y)^4 (2xy^2)^3$ (b) $\tfrac{(x^3)^4 (x^{-2})^5}{(x^6)^5}$ (c) $\left(\tfrac{2xy^2}{x^3 y^{-2}}\right)^2 \left(\tfrac{12xy^3}{x^3 y^{-1}}\right)^{-1}$.

(a)

(b)

(c)

Use scientific notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers $4{,}000$ and $0.004$.

Using place value, we can rewrite the numbers $4{,}000$ and $0.004$. We know that $4{,}000$ means $4 \times 1{,}000$ and $0.004$ means $4 \times \tfrac{1}{1{,}000}$. If we write the $1{,}000$ as a power of ten in exponential form, we can rewrite these numbers in this way:

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 ten, and the second factor is a power of $10$ written in exponential form, it is said to be in scientific notation.

It is customary in scientific notation to use the $\times$ multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation. In both cases, the decimal was moved $3$ places to get the first factor between $1$ and $10$. The power of $10$ is positive when the number is larger than $1$: $4{,}000 = 4 \times 10^3$. The power of $10$ is negative when the number is between $0$ and $1$: $0.004 = 4 \times 10^{-3}$.

Example. Write in scientific notation: (a) $37{,}000$ (b) $0.0052$.

(a) The original number, $37{,}000$, is greater than $1$ so we will have a positive power of $10$. Move the decimal point to get $3.7$, a number between $1$ and $10$. The decimal point was moved $4$ places to the left, so $37{,}000 = 3.7 \times 10^4$. Check: $3.7 \times 10^4 = 3.7 \times 10{,}000 = 37{,}000$. ✓

(b) The original number, $0.0052$, is between $0$ and $1$ so we will have a negative power of $10$. Move the decimal point to get $5.2$, a number between $1$ and $10$. The decimal point was moved $3$ places to the right, so $0.0052 = 5.2 \times 10^{-3}$. Check: $5.2 \times 10^{-3} = 5.2 \times \tfrac{1}{1000} = 5.2 \times 0.001 = 0.0052$. ✓

How can we convert from scientific notation to decimal form? If we look at two numbers written in scientific notation, $9.12 \times 10^4$ and $9.12 \times 10^{-4}$, we see that $9.12 \times 10^4 = 9.12 \times 10{,}000 = 91{,}200$ and $9.12 \times 10^{-4} = 9.12 \times 0.0001 = 0.000912$. In both cases the decimal point moved $4$ places. When the exponent was positive, the decimal moved to the right; when the exponent was negative, the decimal point moved to the left.

Example. Convert to decimal form: (a) $6.2 \times 10^3$ (b) $-8.9 \times 10^{-2}$.

(a) The exponent is $3$. Since the exponent is positive, move the decimal point $3$ places to the right, adding zeros as needed for placeholders: $6.2 \times 10^3 = 6{,}200$.

(b) The exponent is $-2$. Since the exponent is negative, move the decimal point $2$ places to the left, adding zeros as needed for placeholders: $-8.9 \times 10^{-2} = -0.089$.

When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Example. Multiply or divide as indicated. Write answers in decimal form: (a) $(-4 \times 10^5)(2 \times 10^{-7})$ (b) $\tfrac{9 \times 10^3}{3 \times 10^{-2}}$.

(a)

(b)

Key terms

Product Property for Exponents — $a^m \cdot a^n = a^{m+n}$; to multiply with like bases, add the exponents. Quotient Property for Exponents — $\tfrac{a^m}{a^n} = a^{m-n}$ for $a \neq 0$; subtract the exponents. Zero Exponent Property — any non-zero number raised to the zero power is $1$. negative exponent — $a^{-n} = \tfrac{1}{a^n}$ for $a \neq 0$. Power Property for Exponents — $(a^m)^n = a^{m \cdot n}$; to raise a power to a power, multiply the exponents. Product to a Power Property — $(ab)^m = a^m b^m$. Quotient to a Power Property — $\left(\tfrac{a}{b}\right)^m = \tfrac{a^m}{b^m}$. scientific notation — a number written in the form $a \times 10^n$, where $1 \le |a| < 10$ and $n$ is an integer.

This section is adapted from Intermediate Algebra 2e, Section 5.2: Properties of Exponents and Scientific Notation by Lynn Marecek and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the Be Prepared quiz, media links, and end-of-section exercises; recreated the summary of exponent properties as a markdown table; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.