Properties of Exponents and Scientific Notation

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression

a^{m},

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.

| {: valign=”top”}{: .unnumbered .unstyled summary=”The figure shows how to multiply exponentials with the same base. In the example we start with x raised to the power of 2 times x raised to the power of 3. This means the we are multiplying 2 factors of x with 3 factors of x for a total of 5 factors of x so the simplified result is x raised to the power of 5.” data-label=””}

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.

| {: valign=”top”}{: .unnumbered .unstyled summary=”The figure shows two examples of simplifying with the exponent property of division. In the first example the expression is x to the power of 5 divided by x to the power of 2. This means we have 5 factors of x divided by 2 factors of x. Using the equivalent fractions property, we can cross off two factors of x from the numerator and two from the denominator just leaving 3 of the original 5 factors in the numerator. So the simplified expression is x to the power of 3. In the second example the expression is x to the power of 2 divided by x to the power of 3. This means we have 2 factors of x divided by 3 factors of x. Using the equivalent fractions property, we can cross off two factors of x from the numerator and two from the denominator just leaving 1 factor of x in the denominator. So the simplified expression is 1 divided by x.” data-label=””}

Notice, in each case the bases were the same and we subtracted exponents. We see

\frac{x^{5}}{x^{2}}

When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator–notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator.

\frac{x}{x} = 1

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like

\frac{a^{m}}{a^{m}} .

The Quotient Property for Exponents shows us how to simplify

\frac{a^{m}}{a^{m}} .

in two ways to lead us to the definition of the Zero Exponent Property. In general, for

a \neq 0 :

$In the first way we write a to the power of m divided by a to the power of m as a to the power of the quantity m minus m. This is equal to a to the power of 0. In the second way we write a to the power of m divided by a to the power of m as a fraction with m factors of a in the numerator and a factors of m in the denominator. Simplifying this we can cross of all the factors and are left with the number 1. This shows that a to the power of 0 is equal to 1.$ We see

\frac{a^{m}}{a^{m}}

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

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?

We subtract the exponent in the denominator from the exponent in the numerator. We see

\frac{x^{2}}{x^{5}}

$In the figure the expression x raised to the power of 2 divided by x raised to the power of 5 is written as a fraction with 2 factors of x in the numerator divided by 5 factors of x in the denominator. Two factors are crossed off in both the numerator and denominator. This only leaves 3 factors of x in the denominator. The simplified fraction is 1 divided by x to the power of 3.$ This implies that

x^{−3} = \frac{1}{x^{3}}

and it leads us to the definition of a negative exponent. If n is an integer and

a \neq 0,

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.

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.

For example, if after simplifying an expression we end up with the expression

x^{−3},

The answer is considered to be in simplest form when it has only positive exponents.

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.

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.

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

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

| How many factors altogether? |

| {: valign=”top”}| So we have |

| {: valign=”top”}{: .unnumbered .unstyled summary=”The quantity x raised to the power of 2 raised to the power of 3 is written as x to the power of 2 times x to the power of 2 times x to the power of 2. Since each x to the power of 2 is 2 factors of x this is 6 factors of x so we have x to the power of 6.” data-label=””}

Notice the 6 is the product of the exponents, 2 and 3. We see that

{(x^{2})}^{3}

We multiplied the exponents. This leads to the Power Property for Exponents.

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Now we will look at an example that will lead us to the Quotient to a Power Property.

We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties.

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

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.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

The figure shows two examples of converting from standard notation to scientific notation. In one example 4000 is converted to 4 times 10 to the power of 3. The decimal point in 4000 starts at the right and moves 3 places to the left to make the number 4. The 3 places moved make the exponent 3. In the other example, the number 0.004 is converted to 4 times 10 to the negative 3 power. The decimal point in 0.004 is moved 3 places to the right to make the number 4. The 3 places moved make the exponent negative 3.

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}

How can we convert from scientific notation to decimal form? 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.

The figure shows two examples of converting from scientific notation to standard notation. In one example 9.12 times 10 to the power of 4 is converted to 91200. The decimal point in 9.12 moves 4 places to the right to make the number 91200. In the other example, the number 9.12 times 10 to the power of -4 is converted to 0.000912. The decimal point in 9.12 is moved 4 places to the left to make the number 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.

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.

Key Concepts

To raise a fraction to a power, raise the numerator and denominator to that power.

| {: valign=”top”}{: .unnumbered summary=”This table has two columns, labeled Property and Description. The Product Property states that a to the m times a to the n equals a to the m plus n. The Power Property states that a to the m raised to a power of n is a to the m times n. The Quotient Property states that a to the m divided by a to the n equals a to the m minus n when a is not equal to 0. The Zero Exponent Property states that a to the 0 power equals 1 when a is not equal to 0. The Properties of Negative Exponents states that a to the negative n equals 1 over a to the n and 1 over a to the negative n equals a to the n. The Quotient to a Negative Exponent states that a over b all raised to the negative n equals b over a all raised to the n.”}

Practice Makes Perfect

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

ⓐ $d^{3} \cdot d^{6}$

ⓑ $4^{5 x} \cdot 4^{9 x}$

ⓓ $w \cdot w^{2} \cdot w^{3}$

ⓐ $d^{9}$

ⓑ $4^{14 x}$

ⓓ $w^{6}$

ⓐ $x^{4} \cdot x^{2}$

ⓑ $8^{9 x} \cdot 8^{3}$

ⓓ $y \cdot y^{3} \cdot y^{5}$

ⓐ $n^{19} \cdot n^{12}$

ⓑ $3^{x} \cdot 3^{6}$

ⓓ $a^{4} \cdot a^{3} \cdot a^{9}$

ⓐ $n^{31}$

ⓑ $3^{x + 6}$

ⓓ $a^{16}$

ⓐ $q^{27} \cdot q^{15}$

ⓑ $5^{x} \cdot 5^{4 x}$

ⓓ $c^{5} \cdot c^{11} \cdot c^{2}$

m^{x} \cdot m^{3}

m^{x + 3}

n^{y} \cdot n^{2}

y^{a} \cdot y^{b}

y^{a + b}

x^{p} \cdot x^{q}

ⓐ $\frac{x^{18}}{x^{3}}$

ⓑ $\frac{5^{12}}{5^{3}}$

ⓓ $\frac{10^{2}}{10^{3}}$

ⓐ $x^{15}$

ⓑ $5^{9}$

ⓓ $\frac{1}{10}$

ⓐ $\frac{y^{20}}{y^{10}}$

ⓑ $\frac{7^{16}}{7^{2}}$

ⓓ $\frac{8^{3}}{8^{5}}$

ⓐ $\frac{p^{21}}{p^{7}}$

ⓑ $\frac{4^{16}}{4^{4}}$

ⓓ $\frac{4}{4^{6}}$

ⓐ $p^{14}$

ⓑ $4^{12}$

ⓓ $\frac{1}{4^{5}}$

ⓐ $\frac{u^{24}}{u^{3}}$

ⓑ $\frac{9^{15}}{9^{5}}$

ⓓ $\frac{10}{10^{3}}$

ⓐ $20^{0}$

ⓑ $b^{0}$

ⓐ 1 ⓑ 1

ⓐ $13^{0}$

ⓑ $k^{0}$

ⓐ $− 27^{0}$

ⓑ $− (27^{0})$

ⓐ $−1$

ⓑ $−1$

ⓐ $− 15^{0}$

ⓑ $− (15^{0})$

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

ⓐ $a^{−2}$

ⓑ $10^{−3}$

ⓓ $\frac{1}{3^{−2}}$

ⓐ $\frac{1}{a^{2}}$

ⓑ $\frac{1}{1000}$

ⓓ $9$

ⓐ $b^{−4}$

ⓑ $10^{−2}$

ⓓ $\frac{1}{5^{−2}}$

ⓐ $r^{−3}$

ⓑ $10^{−5}$

ⓓ $\frac{1}{10^{−3}}$

ⓐ $\frac{1}{r^{3}}$

ⓑ $\frac{1}{100,000}$

ⓓ $1,000$

ⓐ $s^{−8}$

ⓑ $10^{−2}$

ⓓ $\frac{1}{10^{−4}}$

ⓐ ${(\frac{5}{8})}^{−2}$

ⓑ ${(- \frac{b}{a})}^{−2}$

ⓐ $\frac{64}{25}$

ⓑ $\frac{a^{2}}{b^{2}}$

ⓐ ${(\frac{3}{10})}^{−2}$

ⓑ ${(- \frac{2}{z})}^{−3}$

ⓐ ${(\frac{4}{9})}^{−3}$

ⓑ ${(- \frac{u}{v})}^{−5}$

ⓐ $\frac{729}{64}$

ⓑ $- \frac{v^{5}}{u^{5}}$

ⓐ ${(\frac{7}{2})}^{−3}$

ⓑ ${(- \frac{3}{x})}^{−3}$

ⓐ ${(−5)}^{−2}$

ⓑ $− 5^{−2}$

ⓓ $− {(\frac{1}{5})}^{−2}$

ⓐ $\frac{1}{25}$

ⓑ $\frac{1}{25}$

ⓓ $−25$

ⓐ $− 5^{−3}$

ⓑ ${(- \frac{1}{5})}^{−3}$

ⓓ ${(−5)}^{−3}$

ⓐ $3 \cdot 5^{−1}$

ⓑ ${(3 \cdot 5)}^{−1}$

ⓐ $\frac{3}{5}$

ⓑ $\frac{1}{15}$

ⓐ $3 \cdot 4^{−2}$

ⓑ ${(3 \cdot 4)}^{−2}$

In the following exercises, simplify each expression using the Product Property.

ⓐ $b^{4} b^{−8}$

ⓑ $(w^{4} x^{−5}) (w^{−2} x^{−4})$

ⓐ $\frac{1}{b^{4}}$

ⓑ $\frac{w^{2}}{x^{9}}$

ⓐ $s^{3} \cdot s^{−7}$

ⓑ $(m^{3} n^{−3}) (m^{−5} n^{−1})$

ⓐ $a^{3} \cdot a^{−3}$

ⓑ $(u v^{−2}) (u^{−5} v^{−3})$

ⓐ 1 ⓑ $\frac{1}{u^{4} v^{5}}$

ⓐ $y^{5} \cdot y^{−5}$

ⓑ $(p q^{−4}) (p^{−6} q^{−3})$

p^{5} \cdot p^{−2} \cdot p^{−4}

\frac{1}{p}

x^{4} \cdot x^{−2} \cdot x^{−3}

In the following exercises, simplify each expression using the Power Property.

ⓐ ${(m^{4})}^{2}$

ⓑ ${(10^{3})}^{6}$

ⓐ $m^{8}$

ⓑ $10^{18}$

ⓐ ${(b^{2})}^{7}$

ⓑ ${(3^{8})}^{2}$

ⓐ ${(y^{3})}^{x}$

ⓑ ${(5^{x})}^{y}$

ⓐ $y^{3 x}$

ⓑ $5^{x y}$

ⓐ ${(x^{2})}^{y}$

ⓑ ${(7^{a})}^{b}$

In the following exercises, simplify each expression using the Product to a Power Property.

ⓐ ${(−3 x y)}^{2}$

ⓑ ${(6 a)}^{0}$

ⓓ ${(−4 y^{−3})}^{2}$

ⓐ $9 x^{2} y^{2}$

ⓑ 1 ⓒ $\frac{1}{25 x^{4}}$

ⓓ $\frac{16}{y^{6}}$

ⓐ ${(−4 a b)}^{2}$

ⓑ ${(5 x)}^{0}$

ⓓ ${(−7 y^{−3})}^{2}$

ⓐ ${(−5 a b)}^{3}$

ⓑ ${(−4 p q)}^{0}$

ⓓ ${(3 y^{−4})}^{2}$

ⓐ $−125 a^{3} b^{3}$

ⓑ 1 ⓒ $\frac{1}{36 x^{6}}$

ⓓ $\frac{9}{y^{8}}$

ⓐ ${(−3 x y z)}^{4}$

ⓑ ${(−7 m n)}^{0}$

ⓓ ${(2 y^{−5})}^{2}$

In the following exercises, simplify each expression using the Quotient to a Power Property.

ⓐ ${(\frac{p}{2})}^{5}$

ⓑ ${(\frac{x}{y})}^{−6}$

ⓓ ${(\frac{4 p^{−3}}{q^{2}})}^{2}$

ⓐ $\frac{p^{5}}{32}$

ⓑ $\frac{x^{6}}{y^{6}}$

ⓓ $\frac{16}{p^{6} q^{4}}$

ⓐ ${(\frac{x}{3})}^{4}$

ⓑ ${(\frac{a}{b})}^{−5}$

ⓓ ${(\frac{4 p^{−3}}{q^{2}})}^{2}$

ⓐ ${(\frac{a}{3 b})}^{4}$

ⓑ ${(\frac{5}{4 m})}^{−2}$

ⓓ ${(\frac{4 p^{−3}}{q^{2}})}^{2}$

ⓐ $\frac{a^{4}}{81 b^{4}}$

ⓑ $\frac{16 m^{2}}{25}$

ⓓ $\frac{16}{p^{6} q^{4}}$

ⓐ ${(\frac{x}{2 y})}^{3}$

ⓑ ${(\frac{10}{3 q})}^{−4}$

ⓓ ${(\frac{4 p^{−3}}{q^{2}})}^{2}$

In the following exercises, simplify each expression by applying several properties.

ⓐ ${(5 t^{2})}^{3} {(3 t)}^{2}$

ⓑ $\frac{{(t^{2})}^{5} {(t^{−4})}^{2}}{{(t^{3})}^{7}}$

ⓐ $1125 t^{8}$

ⓑ $\frac{1}{t^{19}}$

ⓐ ${(10 k^{4})}^{3} {(5 k^{6})}^{2}$

ⓑ $\frac{{(q^{3})}^{6} {(q^{−2})}^{3}}{{(q^{4})}^{8}}$

ⓐ ${(m^{2} n)}^{2} {(2 m n^{5})}^{4}$

ⓑ $\frac{{(−2 p^{−2})}^{4} {(3 p^{4})}^{2}}{{(−6 p^{3})}^{2}}$

ⓐ $16 m^{8} n^{22}$

ⓑ $\frac{4}{p^{6}}$

ⓐ ${(3 p q^{4})}^{2} {(6 p^{6} q)}^{2}$

ⓑ $\frac{{(−2 k^{−3})}^{2} {(6 k^{2})}^{4}}{{(9 k^{4})}^{2}}$

Mixed Practice

In the following exercises, simplify each expression.

ⓐ $7 n^{−1}$

ⓑ ${(7 n)}^{−1}$

ⓐ $\frac{7}{n}$

ⓑ $\frac{1}{7 n}$

ⓐ $6 r^{−1}$

ⓑ ${(6 r)}^{−1}$

ⓐ ${(3 p)}^{−2}$

ⓑ $3 p^{−2}$

ⓐ $\frac{1}{9 p^{2}}$

ⓑ $\frac{3}{p^{2}}$

ⓐ ${(2 q)}^{−4}$

ⓑ $2 q^{−4}$

{(x^{2})}^{4} \cdot {(x^{3})}^{2}

x^{14}

{(y^{4})}^{3} \cdot {(y^{5})}^{2}

{(a^{2})}^{6} \cdot {(a^{3})}^{8}

x^{30}

{(b^{7})}^{5} \cdot {(b^{2})}^{6}

{(2 m^{6})}^{3}

2 m^{18}

{(3 y^{2})}^{4}

{(10 x^{2} y)}^{3}

1,000 x^{6} y^{3}

{(2 m n^{4})}^{5}

{(−2 a^{3} b^{2})}^{4}

16 a^{12} b^{8}

{(−10 u^{2} v^{4})}^{3}

{(\frac{2}{3} x^{2} y)}^{3}

\frac{8}{27} x^{6} y^{3}

{(\frac{7}{9} p q^{4})}^{2}

{(8 a^{3})}^{2} {(2 a)}^{4}

1,024 a^{10}

{(5 r^{2})}^{3} {(3 r)}^{2}

{(10 p^{4})}^{3} {(5 p^{6})}^{2}

25,000 p^{24}

{(4 x^{3})}^{3} {(2 x^{5})}^{4}

{(\frac{1}{2} x^{2} y^{3})}^{4} {(4 x^{5} y^{3})}^{2}

x^{18} y^{18}

{(\frac{1}{3} m^{3} n^{2})}^{4} {(9 m^{8} n^{3})}^{2}

{(3 m^{2} n)}^{2} {(2 m n^{5})}^{4}

144 m^{8} n^{22}

{(2 p q^{4})}^{3} {(5 p^{6} q)}^{2}

ⓐ ${(3 x)}^{2} (5 x)$

ⓑ ${(2 y)}^{3} (6 y)$

ⓐ $45 x^{3}$

ⓑ $48 y^{4}$

ⓐ ${(\frac{1}{2} y^{2})}^{3} {(\frac{2}{3} y)}^{2}$

ⓑ ${(\frac{1}{2} j^{2})}^{5} {(\frac{2}{5} j^{3})}^{2}$

ⓐ ${(2 r^{−2})}^{3} {(4^{−1} r)}^{2}$

ⓑ ${(3 x^{−3})}^{3} {(3^{−1} x^{5})}^{4}$

ⓐ $\frac{1}{2 r^{4}}$

ⓑ $\frac{1}{3} x^{11}$

{(\frac{k^{−2} k^{8}}{k^{3}})}^{2}

{(\frac{j^{−2} j^{5}}{j^{4}})}^{3}

\frac{1}{j^{3}}

\frac{{(−4 m^{−3})}^{2} {(5 m^{4})}^{3}}{{(−10 m^{6})}^{3}}

\frac{{(−10 n^{−2})}^{3} {(4 n^{5})}^{2}}{{(2 n^{8})}^{2}}

- \frac{4000}{n^{12}}

Use Scientific Notation

In the following exercises, write each number in scientific notation.

ⓐ 57,000 ⓑ 0.026

ⓐ 340,000 ⓑ 0.041

ⓐ $34 \times 10^{4}$

ⓑ $41 \times 10^{−3}$

ⓐ 8,750,000 ⓑ 0.00000871

ⓐ 1,290,000 ⓑ 0.00000103

ⓐ $1.29 \times 10^{6}$

ⓑ $103 \times 10^{−8}$

In the following exercises, convert each number to decimal form.

ⓐ $5.2 \times 10^{2}$

ⓑ $2.5 \times 10^{−2}$

ⓐ $−8.3 \times 10^{2}$

ⓑ $3.8 \times 10^{−2}$

ⓐ $−830$

ⓑ 0.038

ⓐ $7.5 \times 10^{6}$

ⓑ $−4.13 \times 10^{−5}$

ⓐ $1.6 \times 10^{10}$

ⓑ $8.43 \times 10^{−6}$

ⓐ 16,000,000,000* * *

ⓑ 0.00000843

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

ⓐ $(3 \times 10^{−5}) (3 \times 10^{9})$

ⓑ $\frac{7 \times 10^{−3}}{1 \times 10^{−7}}$

ⓐ $(2 \times 10^{2}) (1 \times 10^{−4})$

ⓑ $\frac{5 \times 10^{−2}}{1 \times 10^{−10}}$

ⓐ 0.02 ⓑ 500,000,000

ⓐ $(7.1 \times 10^{−2}) (2.4 \times 10^{−4})$

ⓑ $\frac{6 \times 10^{4}}{3 \times 10^{−2}}$

ⓐ $(3.5 \times 10^{−4}) (1.6 \times 10^{−2})$

ⓑ $\frac{8 \times 10^{6}}{4 \times 10^{−1}}$

ⓐ 0.0000056 ⓑ 20,000,000

Writing Exercises

Use the Product Property for Exponents to explain why $x \cdot x = x^{2} .$

Jennifer thinks the quotient $\frac{a^{24}}{a^{6}}$

simplifies to $a^{4} .$

What is wrong with her reasoning?

Answers will vary.

Explain why $− 5^{3} = {(−5)}^{3}$

but $− 5^{4} \neq {(−5)}^{4} .$

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all goals?

The original number, 37,000, is greater than 1 so we will have a positive power of 10.	37,000
Move the decimal point to get 3.7, a number between 1 and 10.
Count the number of decimal places the point was moved.
Write as a product with a power of 10.
Check: $\begin{matrix} 3.7 \times 10^{4} \\ 3.7 \times 10,000 \\ 37,000 \end{matrix}$

The original number, 0.0052, is between 0 and 1 so we will have a negative power of 10.	0.0052
Move the decimal point to get 5.2, a number between 1 and 10.
Count the number of decimal places the point was moved.
Write as a product with a power of 10.
$\begin{matrix} Check: & 5.2 \times 10^{−3} \\ 5.2 \times \frac{1}{10^{3}} \\ 5.2 \times \frac{1}{1000} \\ 5.2 \times 0.001 \\ 0.0052 \end{matrix}$


Determine the exponent, n, on the factor 10.
	The exponent is 3.
Since the exponent is positive, move the decimal point 3 places to the right.
Add zeros as needed for placeholders.


Determine the exponent, n, on the factor 10.	The exponent is $−2 .$
Since the exponent is negative, move the decimal point 2 places to the left.
Add zeros as needed for placeholders.


{: valign=”top”}	Use the Product Property, $a^{m} \cdot a^{n} = a^{m + n} .$


Rewrite, $a = a^{1} .$
Use the Commutative Property and use the Product Property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.


{: valign=”top”}	What do they mean?	$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$


{: valign=”top”}	Use the Equivalent Fractions Property.	$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$


{: valign=”top”}	Use Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n} .$

in the denominator.
{: valign=”top”}	Use Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}} .$


{: valign=”top”}	Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$


{: valign=”top”}	Use Power of a Product Property, ${(a b)}^{m} = a^{m} b^{m} .$


{: valign=”top”}	Use Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}} .$

Property	Description
{: valign=”top”}	———-
Product Property	$a^{m} \cdot a^{n} = a^{m + n}$


{: valign=”top”}	Power Property	${(a^{m})}^{n} = a^{m \cdot n}$


{: valign=”top”}	Product to a Power	${(a b)}^{n} = a^{m} b^{m}$


{: valign=”top”}	Quotient Property	$\frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0$


{: valign=”top”}	Zero Exponent Property	$a^{0} = 1, a \neq 0$


{: valign=”top”}	Quotient to a Power Property	${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0$


{: valign=”top”}	Properties of Negative Exponents	$a^{− n} = \frac{1}{a^{n}}$


{: valign=”top”}	Quotient to a Negative Exponent	${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$


{: valign=”top”}	Quotient to a Power Property:	${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0$

Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

Use the Definition of a Negative Exponent

Use Scientific Notation

Key Concepts

Practice Makes Perfect

Mixed Practice

Writing Exercises

Self Check

Glossary