# 5.2 Properties of Exponents and Scientific Notation

Topics covered in this section are:

## 5.2.1 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.

### EXPONENTIAL NOTATION

This is read $a$ to the $m^{th}$ power.

In the expression $a^{m}$, the exponent $m$ tells us how many times we use the base $a$ as a factor.

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.

$x^{2} \cdot x^{3}$
$\underbrace{x \cdot x} \cdot \underbrace{x\cdot x \cdot x}$
$\underbrace{2 \text{ factors} \ \ \ \ 3 \text{ factors}}$
$\ \ \ \ \ \ 5 \text{ factors} \ \ \ \ \ \ \$
$x^{5}$

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.

### PRODUCT PROPERTY FOR EXPONENTS

If $a$ is a real number and $m$ and $n$ are integers, then

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

To multiply with like bases, add the exponents.

#### Example 1

Simplify each expression:

• $y^{5} \cdot y^{6}$
• $2^{x} \cdot 2^{3x}$
• $2a^{7} \cdot 3a$
• $d^{4} \cdot d^{5} \cdot d^{2}$
Solution

Part 1

Part 2

Part 3

Part 4

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

Notice, in each case the bases were the same and we subtracted exponents. We see $\frac{x^{5}}{x^{2}}$ is $x^{5−2}$ or $x^{3}$. We see $\frac{x^{2}}{x^{3}}$ is $\frac{1}{x^{3-2}}$ or $\frac{1}{x}$. 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. $\frac{\cancel{x}}{\cancel{x}}=1$. This leads to the Quotient Property for Exponents.

### QUOTIENT PROPERTY FOR EXPONENTS

If $a$ is a real number, $a≠0$, and $m$ and $n$ are integers, then

$\frac{a^{m}}{a^{n}} = a^{m-n}, m>n \text{ and } \frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}, n>m$

#### Example 2

Simplify each expression:

• $\frac{x^{9}}{x^{7}}$
• $\frac{3^{10}}{3^{2}}$
• $\frac{b^{8}}{b^{12}}$
• $\frac{7^{3}}{7^{5}}$
Solution

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

Part 1

Part 2

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

Part 3

Part 4

Notice that when the larger exponent is in the denominator, we are left with factors 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 $\frac{a^{m}}{a^{m}}$. We know,$\frac{x}{x}=1$, for any $x (x≠0)$ since any number divided by itself is $1$.

The Quotient Property for Exponents shows us how to simplify $\frac{a^{m}}{a^{m}}$, when $m>n$ and when $n<m$ by subtracting exponents. What if $m=n$? We will simplify $\frac{a^{m}}{a^{m}}$ in two ways to lead us to the definition of the Zero Exponent Property. In general, for $a≠0$:

We see $\frac{a^{m}}{a^{n}}$ simplifies to $a^{0}$ and to $1$. So $a^{0}=1$. Any non-zero base raised to the power of zero equals $1$.

### ZERO EXPONENT PROPERTY

If $a$ is a non-zero number, then $a^{0}=1$.
If $a$ is a non-zero number, then $a$ to the power of zero equals $1$.
Any non-zero number raised to the power is $1$.

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

#### Example 3

Simplify each expression:

• $9^{0}$
• $n^{0}$
Solution

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

 Part 1 $9^{0}$ Use the definition of the zero exponent. $1$ Part 2 $n^{0}$ Use the definition of the zero exponent. $1$

To simplify the expression $n$ raised to the zero power we just use the definition of the zero exponent. The result is $1$.

## 5.2.2 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 $\frac{x^{2}}{x^{5}}$. We subtract the exponent in the denominator from the exponent in the numerator. We see $\frac{x^{2}}{x^{5}}$ is $x^{2−5}$ or $x^{−3}$.

We can also simplify $\frac{x^{2}}{x^{5}}$ by dividing out common factors:

$\frac{x^{2}}{x^{5}}$
$\frac{ \cancel{x} \cdot \cancel{x}} {\cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot x}$
$\frac{1}{x^{3}}$

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 $a≠0$, then $a^{-n}=\frac{1}{a^{n}}$.

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

### PROPERTIES OF NEGATIVE EXPONENTS

If $n$ is an integer and $a≠0$, then $a^{-n}=\frac{1}{a^{n}}$ or $\frac{1}{a^{-n}}=a^{n}$.

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 $x^{−3}$, we will take one more step and write $\frac{1}{x^{3}}$. The answer is considered to be in simplest form when it has only positive exponents.

#### Example 4

Simplify each expression:

• $x^{-5}$
• $10^{-3}$
• $\frac{1}{y^{-4}}$
• $\frac{1}{3^{-2}}$
Solution

Part 1

Part 2

Part 3

Part 4

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.

This leads us to the Quotient to a Negative Power Property.

### QUOTIENT TO A NEGATIVE POWER PROPERTY

If $a$ and $b$ are real numbers, $a≠0$, $b≠0$ and $n$ is an integer, then

$\left( \frac{a}{b} \right) ^{-n}= \left( \frac{b}{a} \right) ^{n}$

#### Example 5

Simplify each expression:

• $(\frac{5}{7})^{-2}$
• $(-\frac{x}{y})^{-3}$
Solution

Part 1

Part 2

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

#### Example 6

Simplify each expression:

• $z^{-5} \cdot z^{-3}$
• $(m^{4}n^{-3})(m^{-5}n^{-2})$
• $(2x^{-6}y^{8})(-5x^{5}y^{-3})$
Solution

Part 1

Part 2

Part 3

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

 $(x^{2})^{3}$ What does this mean? $x^{2} \cdot x^{2} \cdot x^{2}$ How many factors altogether? So we have $x^{6}$

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 multiplies the exponents. This leads to the Power Property For Exponents.

### POWER PROPERTY FOR EXPONENTS

If $a$ is a real number and $m$ and $n$ are integers, then

$(a^{m})^{n}=a^{m \cdot n}$

To raise a power to a power, multiply the exponents.

#### Example 7

Simplify each expression:

• $(y^{5})^{9}$
• $(4^{4})^{7}$
• $(y^{3})^{6}(y^{5})^{4}$
Solution

Part 1

Part 2

Part 3

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

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

### PRODUCT TO A POWER PROPERTY FOR EXPONENTS

If $a$ and $b$ are real numbers and $m$ is a whole number, then

$(ab)^{m}=a^{m}b^{m}$

To raise a product to a power, raise each factor to that power.

#### Example 8

Simplify each expression:

• $(-3mn)^{3}$
• $(-4a^{2}b)^{0}$
• $(6k^{3})^{-2}$
• $(5x^{-3})^{2}$
Solution

Part 1

Part 2

Part 3

Part 4

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

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

### QUOTIENT TO A POWER PROPERTY FOR EXPONENTS

If $a$ and $b$ are real numbers, $b≠0$, and $m$ is an integer, then

$(\frac{a}{b})^{m} = \frac{a^{m}}{b^{m}}$

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

#### Example 9

Simplify each expression:

• $(\frac{b}{3})^{4}$
• $(\frac{k}{j})^{-3}$
• $(\frac{2xy^{2}}{z})^{3}$
• $(\frac{4p^{-3}}{q^{2}})^{2}$
Solution

Part 1

Part 2

Part 3

Part 4

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.

### SUMMARY OF EXPONENT PROPERTIES

If $a$ and $b$ are real numbers, and $m$ and $n$ are integers, then

#### Example 10

Simplify each expression by applying several properties:

• $(3x^{2}y)^{4}(2xy^{2})^{3}$
• $\frac{(x^{3})^{4}(x^{-2})^{5}}{(x^{6})^{5}}$
• $\left( \frac{2xy^{2}}{x^{3}y^{-2}} \right) ^{2} \left( \frac{12xy^{3}}{x^{3}y^{-1}} \right) ^{-1}$
Solution

Part 1

Part 2

Part 3

## 5.2.3 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 $4 \times \frac{1}{1,000}$.

If we rewrite 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.

### SCIENTIFIC NOTATION

A number is expressed in scientific notation when it is of the form

$a \times 10^{n} \text{ where } 1≤ |a|<10$ and $n$ is an integer.

It is customary in scientific notation to use as 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 $1$: $0.004 =4 \times 10^{-3}$.

### HOW TO: To convert a decimal to scientific notation.

1. Move the decimal point so that the first factor is greater than or equal to $1$ but less than $10$.
2. Count the number of decimal places, $n$, that the decimal point was moved.
3. Write the number as a product with a power of $10$. If the original number is.
• greater than $1$, the power of $10$ will be $10^{n}$.
• between $0$ and $1$, the power of $10$ will be $10^{−n}$.
4. Check.

#### Example 11

Write in scientific notation:

• $37,000$
• $0.0052$
Solution

Part 1

Part 2

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

 $9.12 \times 10^{4}$ $9.12 \times 10^{-4}$ $9.12 \times 10,000$ $9.12 \times 0.0001$ $91,200$ $0.000912$

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.

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.

### HOW TO: Convert scientific notation to decimal form.

1. Determine the exponent, $n$, on the factor $10$.
2. Move the decimal $n$ places, adding zeros if needed.
• If the exponent is positive, move the decimal point $n$ places to the right.
• If the exponent is negative, move the decimal point $|n|$ places to the left.
3. Check.

#### Example 12

Convert to decimal form:

• $6.2 \times 10^{3}$
• $-8.9 \times 10^{-2}$
Solution

Part 1

Part 2

When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a w/ay 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 13

Multiply or divide as indicated. Write answers in decimal form:

• $(-4 \times 10^{5})(2 \times 10^{-7})$
• $\frac{9 \times 10^{3}}{3 \times 10^{-2}}$
Solution

Part 1

Part 2