Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

We saw earlier in this chapter 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 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 us to the definition of 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 an 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 $\tfrac{1}{x^3}$. The answer is considered to be in simplest form when it has only positive exponents.

Example. Simplify: (a) $4^{-2}$ (b) $10^{-3}$.

(a) Use the definition of a negative exponent, $a^{-n} = \tfrac{1}{a^n}$: $4^{-2} = \tfrac{1}{4^2} = \tfrac{1}{16}$.

(b) Use the definition of a negative exponent: $10^{-3} = \tfrac{1}{10^3} = \tfrac{1}{1000}$.

When we raised a fraction whose numerator is one and whose denominator is an integer to a negative exponent, we found:

This leads to the Property of Negative Exponents.

Example. Simplify: (a) $\tfrac{1}{y^{-4}}$ (b) $\tfrac{1}{3^{-2}}$.

(a) Use the Property of Negative Exponents, $\tfrac{1}{a^{-n}} = a^n$: $\tfrac{1}{y^{-4}} = y^4$.

(b) Use the Property of Negative Exponents, then simplify: $\tfrac{1}{3^{-2}} = 3^2 = 9$.

Suppose now we have a fraction raised to a negative exponent. Using the definition of a negative exponent, we can show that taking the reciprocal of the base and changing the sign of the exponent works here too. For example, $\left(\tfrac{3}{4}\right)^{-2} = \left(\tfrac{4}{3}\right)^2$. This leads us to the Quotient to a Negative Exponent Property.

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

(a) Use the Quotient to a Negative Exponent Property — take the reciprocal of the fraction and change the sign of the exponent, then simplify:

(b) Take the reciprocal of the fraction and change the sign of the exponent, then simplify:

When simplifying an expression with exponents, we must be careful to correctly identify the base.

Example. Simplify: (a) $(-3)^{-2}$ (b) $-3^{-2}$ (c) $\left(-\tfrac{1}{3}\right)^{-2}$ (d) $-\left(\tfrac{1}{3}\right)^{-2}$.

(a) Here the exponent applies to the base $-3$. Take the reciprocal of the base and change the sign of the exponent: $(-3)^{-2} = \tfrac{1}{(-3)^2} = \tfrac{1}{9}$.

(b) The expression $-3^{-2}$ means “find the opposite of $3^{-2}$.” Rewrite as a product with $-1$, then apply the negative exponent: $-3^{-2} = -1 \cdot 3^{-2} = -1 \cdot \tfrac{1}{3^2} = -\tfrac{1}{9}$.

(c) Here the exponent applies to the base $-\tfrac{1}{3}$. Take the reciprocal of the base and change the sign of the exponent: $\left(-\tfrac{1}{3}\right)^{-2} = \left(-\tfrac{3}{1}\right)^2 = 9$.

(d) The expression $-\left(\tfrac{1}{3}\right)^{-2}$ means “find the opposite of $\left(\tfrac{1}{3}\right)^{-2}$”: $-\left(\tfrac{1}{3}\right)^{-2} = -1 \cdot \left(\tfrac{3}{1}\right)^2 = -9$.

We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.

Example. Simplify: (a) $4 \cdot 2^{-1}$ (b) $(4 \cdot 2)^{-1}$.

(a) Do exponents before multiplication. Use $a^{-n} = \tfrac{1}{a^n}$, then simplify: $4 \cdot 2^{-1} = 4 \cdot \tfrac{1}{2^1} = 2$.

(b) Simplify inside the parentheses first, then apply the negative exponent: $(4 \cdot 2)^{-1} = (8)^{-1} = \tfrac{1}{8^1} = \tfrac{1}{8}$.

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero.

Example. Simplify: (a) $x^{-6}$ (b) $\left(u^4\right)^{-3}$.

(a) Use the definition of a negative exponent, $a^{-n} = \tfrac{1}{a^n}$: $x^{-6} = \tfrac{1}{x^6}$.

(b) Use the definition of a negative exponent, then simplify: $\left(u^4\right)^{-3} = \tfrac{1}{\left(u^4\right)^3} = \tfrac{1}{u^{12}}$.

When there is a product and an exponent, we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents.

Example. Simplify: (a) $5y^{-1}$ (b) $(5y)^{-1}$ (c) $(-5y)^{-1}$.

(a) Notice the exponent applies to just the base $y$. Take the reciprocal of $y$ and change the sign of the exponent: $5y^{-1} = 5 \cdot \tfrac{1}{y^1} = \tfrac{5}{y}$.

(b) Here the parentheses make the exponent apply to the base $5y$. Take the reciprocal of $5y$ and change the sign of the exponent: $(5y)^{-1} = \tfrac{1}{(5y)^1} = \tfrac{1}{5y}$.

(c) The base here is $-5y$. Take the reciprocal of $-5y$ and change the sign of the exponent, then simplify: $(-5y)^{-1} = \tfrac{1}{(-5y)^1} = \tfrac{1}{-5y} = -\tfrac{1}{5y}$.

With negative exponents, the Quotient Rule needs only one form $\tfrac{a^m}{a^n} = a^{m-n}$, for $a \neq 0$. When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.

Simplify Expressions with Integer Exponents

All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Example. Simplify: (a) $x^{-4} \cdot x^6$ (b) $y^{-6} \cdot y^4$ (c) $z^{-5} \cdot z^{-3}$.

(a) Use the Product Property, $a^m \cdot a^n = a^{m+n}$: $x^{-4} \cdot x^6 = x^{-4+6} = x^2$.

(b) The bases are the same, so add the exponents: $y^{-6} \cdot y^4 = y^{-6+4} = y^{-2} = \tfrac{1}{y^2}$.

(c) Add the exponents, then apply the definition of a negative exponent: $z^{-5} \cdot z^{-3} = z^{-5-3} = z^{-8} = \tfrac{1}{z^8}$.

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.

Example. Simplify: $\left(m^4 n^{-3}\right)\left(m^{-5} n^{-2}\right)$.

Use the Commutative Property to get like bases together, add the exponents for each base, then take reciprocals and change the signs of the exponents:

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

Example. Simplify: $\left(6k^3\right)^{-2}$.

Use the Product to a Power Property, then the Power Property, then the definition of a negative exponent, then simplify:

Example. Simplify: $\left(5x^{-3}\right)^2$.

Use the Product to a Power Property, simplify $5^2$ and multiply the exponents of $x$ using the Power Property, then rewrite $x^{-6}$ with the definition of a negative exponent:

To simplify a fraction, we use the Quotient Property and subtract the exponents.

Example. Simplify: $\tfrac{r^5}{r^{-4}}$.

Use the Quotient Property, $\tfrac{a^m}{a^n} = a^{m-n}$, then simplify:

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of $10$. Consider 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 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 $\times$ as the 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 $4{,}000$, the decimal was moved $3$ places to the left; in $0.004$, it was moved $3$ places to the right. 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 $1$, and negative when the number is between $0$ and $1$.

Example. Write in scientific notation: $37{,}000$.

Move the decimal point after the $3$ so the first factor, $3.7$, is between $1$ and $10$. The decimal point was moved $4$ places to the left. Since $37{,}000$ is greater than $1$, the power of $10$ will have exponent $4$:

Check: $10^4$ is $10{,}000$, and $10{,}000$ times $3.7$ is $37{,}000$. ✓

Example. Write in scientific notation: $0.0052$.

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:

Check: $5.2 \times 10^{-3} = 5.2 \times \tfrac{1}{1000} = 5.2 \times 0.001 = 0.0052$. ✓

Convert Scientific Notation to Decimal Form

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. When the exponent was positive, the decimal moved $4$ places to the right; when the exponent was negative, the decimal moved $4$ places to the left.

Example. Convert to decimal form: $6.2 \times 10^3$.

The exponent is $3$, and it is positive, so move the decimal point $3$ places to the right. We need to add $2$ zeros as placeholders:

Check: $10^3$ is $1000$, and $1000$ times $6.2$ will be $6{,}200$. ✓

Example. Convert to decimal form: $8.9 \times 10^{-2}$.

The exponent is $-2$. Since it is negative, move the decimal point $2$ places to the left, adding zeros as needed for placeholders:

Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. 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. Write answers in decimal form: $\left(4 \times 10^5\right)\left(2 \times 10^{-7}\right)$.

Use the Commutative Property to rearrange the factors, multiply, then change to decimal form:

Example. Divide. Write answers in decimal form: $\tfrac{9 \times 10^3}{3 \times 10^{-2}}$.

Separate the factors, rewriting as the product of two fractions, divide, then change to decimal form:

Key terms

negative exponent — for $a \neq 0$, $a^{-n} = \tfrac{1}{a^n}$; a negative exponent means take the reciprocal of the base and change the sign of the exponent. Property of Negative Exponents — $\tfrac{1}{a^{-n}} = a^n$. Quotient to a Negative Exponent Property — $\left(\tfrac{a}{b}\right)^{-n} = \left(\tfrac{b}{a}\right)^n$. scientific notation — a number written in the form $a \times 10^n$, where $1 \le |a| < 10$ and $n$ is an integer; the power of $10$ is positive for numbers larger than $1$ and negative for numbers between $0$ and $1$.

This section is adapted from Elementary Algebra 2e, Section 6.7: Integer Exponents and Scientific Notation by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the worked-example step tables as prose and typeset equations; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.