Further Exponents

Rules for Exponent Arithmetic

There are rules for operating on numbers with exponents that make it easy to simplify and solve problems.

Learning Objectives

Explain and implement the rules for operating on numbers with exponents

Key Takeaways

Key Points

  • The rule [latex]a^m \cdot a^n = a^{m+n}[/latex] applies when multiplying two exponential expressions with the same base, provided the base is a non-zero integer.
  • The rule [latex]\displaystyle \frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex] applies when dividing two exponential expressions with the same base, provided the base is a non-zero integer.
  • The rule [latex]{({a}^{n})}^{m}={a}^{n \cdot m}[/latex] applies when raising an exponential expression to another exponent for any non-zero integer [latex]a[/latex].
  • The rule [latex]{(ab)}^{n}={a}^{n}{b}^{n}[/latex]applies when raising a product to an exponent for any non-zero integers [latex]a[/latex] and [latex]b[/latex].

Key Terms

  • base: In an exponential expression, the value that is multiplied by itself.
  • exponent: In an exponential expression, the value raised above the base; represents the number of times the base must be multiplied by itself.

There are several useful rules for operating on numbers with exponents. The following four rules, also known as “identities,” hold for all integer exponents, provided that the base is non-zero.

Multiplying Exponential Expressions with the Same Base

[latex]a^m \cdot a^n = a^{m+n}[/latex]

[latex]a^m[/latex] means that you have [latex]m[/latex] factors of [latex]a[/latex]. If you multiply this quantity by [latex]a^n[/latex], i.e. by [latex]n[/latex] additional factors of [latex]a[/latex], then you have [latex]a^{m+n}[/latex] factors in total. For example:

[latex]5^3 \cdot 5^4=5^{3+4}=5^7[/latex]

Note that you can only add exponents in this way if the corresponding terms have the same base.

Dividing Exponential Expressions with the Same Base

[latex]\displaystyle \frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]

In the same way that [latex]{ a }^{ m }\cdot { a }^{ n }={ a }^{ m+n }[/latex] because you are adding on factors of [latex]a[/latex], dividing removes factors of [latex]a[/latex]. If you have [latex]n[/latex] factors of [latex]a[/latex] in the denominator, then you can cross out [latex]n[/latex] factors from the numerator. If there were [latex]m[/latex] factors in the numerator, now you have [latex](m-n)[/latex] factors in the numerator.

In order to visualize this process, consider the fraction:

[latex]\dfrac{3^5}{3^2}[/latex]

This fraction can be rewritten as:

[latex]\dfrac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3}[/latex]

Here you can see that two 3s will cancel out from the numerator and denominator. We are left with:

[latex]\dfrac{3 \cdot 3 \cdot 3}{1} = 3^3[/latex]

As an additional example:

[latex]\displaystyle \frac{7^4}{7^2}=7^{4-2}=7^2[/latex]

Raising an Exponential Expression to an Exponent

[latex]{({a}^{n})}^{m}={a}^{n \cdot m}[/latex]

If you think about an exponent as telling you that you have a certain number of factors of the base, then [latex]{({a}^{n})}^{m}[/latex] means that you have factors [latex]m[/latex] of [latex]a^n[/latex]. Therefore, you have [latex]m[/latex] groups of [latex]a^n[/latex], and each one of those has [latex]n[/latex] groups of [latex]a[/latex]. Therefore, you have [latex]m[/latex] groups of [latex]n[/latex] groups of [latex]a[/latex]; therefore, you have [latex]n \cdot m[/latex] groups of [latex]a[/latex], or [latex]{a}^{n \cdot m}[/latex]. For example:

[latex](3^3)^3=3^{3 \cdot 3}=3^9[/latex]

Raising a Product to an Exponent

[latex]{(ab)}^{n}={a}^{n}{b}^{n}[/latex]

You can multiply numbers in any order you please. Instead of multiplying together [latex]n[/latex] factors equal to [latex]ab[/latex], you could multiply all of the [latex]a[/latex]s together and all of the [latex]b[/latex]s together and then finish by multiplying [latex]a^n[/latex] by [latex]b^n[/latex]. For example:

[latex](4\cdot5)^3=4^3\cdot5^3[/latex]

Example

Simplify the following expression: [latex](3\cdot2)^3\cdot (2^5)^4[/latex]

For the first part of the expression, apply the rule for a product raised to an exponent:

[latex](3\cdot2)^3\cdot (2^5)^4 = 3^3 \cdot 2^3 \cdot (2^5)^4[/latex]

For the last part of the expression, apply the rule for raising an exponential expression to an exponent:

[latex]3^3 \cdot 2^3 \cdot (2^5)^4 = 3^3 \cdot 2^3 \cdot 2^{5\cdot 4} = 3^3 \cdot 2^3 \cdot 2^{20}[/latex]

Notice that two of the terms in this expression have the same base: 2. These two terms can be combined by applying the rule for multiplying exponential expressions with the same base:

[latex]3^3 \cdot 2^3 \cdot 2^{20} = 3^3 \cdot 2^{3+20} = 3^3 \cdot 2^{23}[/latex]

Therefore, [latex]3^3 \cdot 2^{23}[/latex] is the simplified form of this expression.

Negative Exponents

Numbers with negative exponents are treated normally in arithmetic operations and can be rewritten as fractions.

Learning Objectives

Relate negative exponents to fractions and work with them accordingly

Key Takeaways

Key Points

  • An exponential expression with a negative integer in the exponent can be rewritten as a fraction by applying the rule [latex]b^{-n} = \frac{1}{b^n}[/latex].
  • The rules for operating on numbers with exponents still apply when the exponent is a negative integer.

Solving mathematical problems involving negative exponents may seem daunting. However, negative exponents are treated much like positive exponents when applying the rules for operations. There is an additional rule that allows us to change the negative exponent to a positive one in the denominator of a fraction, and it holds true for any real numbers [latex]n[/latex] and [latex]b[/latex], where [latex]b \neq 0[/latex]:

[latex]b^{-n} = \dfrac{1}{b^n}[/latex]

For example:

[latex]6^{-2} = \dfrac{1}{6^2} = \dfrac{1}{36}[/latex]

To understand how this rule is derived, consider the following fraction:

[latex]\dfrac{7^3}{7^5}[/latex]

We can rewrite this as:

[latex]\dfrac{7 \cdot 7 \cdot 7}{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}[/latex]

We then notice that three 7s cancel from both the numerator and denominator, and we are left with:

[latex]\dfrac{1}{7 \cdot 7} = \dfrac{1}{7^2}[/latex]

Note that if we apply the rule for division of numbers with exponents, we have:

[latex]\dfrac{7^3}{7^5} = 7^{(3-5)}= 7^{-2}[/latex]

Thus, we can identify that:

[latex]\dfrac{1}{7^2} = 7^{-2}[/latex]

This rule makes it possible to simplify expressions with negative exponents.

Note that each of the rules for operations on numbers with exponents still apply when the exponent is a negative number. For example, consider the rule for multiplying two exponential expressions with the same base. The following is true:

[latex]3^{-4} \cdot 3^2 = 3^{-4+2} = 3^{-2} = \dfrac{1}{3^2}[/latex]

Example

Simplify the following expression: [latex](2^{-4})^2[/latex].

Note that the rule for raising an exponential expression to another exponent can be applied:

[latex](2^{-4})^2 = 2^{(-4)(2)} = 2^{-8} [/latex]

This can be simplified using the rule for negative exponents:

[latex]2^{-8}=\dfrac{1}{2^8}[/latex]

Example

Simplify the following expression: [latex](3^{-2} \cdot 3^4)^{-3}[/latex].

Recall that the rule for multiplying two exponential expressions with the same base can be applied. Therefore, we can simplify the expression inside the parentheses:

[latex]3^{-2} \cdot 3^4 = 3^{-2+4} = 3^2[/latex]

Now place this value back into the parentheses, and apply the rule for raising an exponential expression to an exponent:

[latex](3^2)^{-3} = 3^{(2)(-3)} = 3^{-6} = \dfrac{1}{3^6}[/latex]

Rational Exponents

Rational exponents are another method for writing radicals and can be used to simplify expressions involving both exponents and roots.

Learning Objectives

Relate rational exponents to radicals and the rules for manipulating them

Key Takeaways

Key Points

  • If [latex]b[/latex] is a positive real number and [latex]n[/latex] is a positive integer, then there is exactly one positive real solution to [latex]x^n = b [/latex]. This solution is called the principal [latex]n[/latex]th root of [latex]b[/latex], denoted [latex]\sqrt[n]{b}[/latex] or [latex]\displaystyle b^{\frac{1}{n}}[/latex].
  • A power of a positive real number [latex]b[/latex] with a rational exponent [latex]\frac{m}{n}[/latex] in lowest terms satisfies [latex]{b}^{\frac {m}{n}}= {({b}^{m})}^{\frac{1}{n}}=\sqrt[n]{{b}^{m}}[/latex].
  • The rule for multiplying numbers with rational exponents is [latex]\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}[/latex].
  • The rule for dividing numbers with rational exponents is [latex]\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}[/latex].
  • Writing an expression in the form [latex]{b}^{\frac {m}{n}}[/latex]can allow you to simplify by cancelling powers and roots.

Key Terms

  • root: A number that when raised to a specified power yields the specified number or expression.
  • rational number: A real number that can be expressed as the ratio of two integers.
  • exponent: The power raised above the base, representing the number of times the base must be multiplied by itself.

A rational exponent is a rational number that provides another method for writing roots. For example, an [latex]n[/latex]th root of a number [latex]b[/latex] is a number [latex]x[/latex] such that [latex]x^n = b[/latex]. If [latex]b[/latex] is a positive real number and [latex]n[/latex] is a positive integer, then there is exactly one positive real solution to [latex] x^n = b[/latex]. This solution is called the [latex]n[/latex]th root of [latex]b[/latex] and is denoted [latex]\sqrt[n]{b}[/latex] or [latex]b^\frac{1}{n}[/latex]. For example: [latex]\sqrt{4}=4^\frac{1}{2}=2[/latex].

There are also cases where the exponent is a fraction [latex]\frac{m}{n}[/latex], where [latex]m[/latex] is an integer and [latex]n[/latex] is a positive integer. In such cases, the exponent acts as both a whole number exponent and a root, or fraction exponent. In other words, the following holds true:

[latex]\displaystyle{{b}^{\frac {m}{n}}= {({b}^{m})}^{\frac{1}{n}}=\sqrt[n]{{b}^{m}}}[/latex]

where [latex]b[/latex] is a real number and the rational exponent [latex]\frac{m}{n}[/latex] is a fraction in lowest terms.

The following rules hold true about the signs of roots and rational exponents. For a rational exponent [latex]\frac{m}{n}[/latex], where [latex]\frac{m}{n}[/latex] is in lowest terms:

  • The root is positive if [latex]m[/latex] is even; for example, [latex](-27)^\frac{2}{3}=9[/latex].
  • The root is negative for negative [latex]b[/latex] if [latex]m[/latex] and [latex]n[/latex] are odd; for example, [latex]\displaystyle (-27)^\frac{1}{3}=-3[/latex].
  • The root can be either sign if [latex]b[/latex] is positive and [latex]n[/latex] is even; for example, [latex]64^\frac{1}{2}[/latex]has two roots: [latex]8[/latex] and [latex]-8[/latex].

Note that since there is no real number [latex]x[/latex] such that [latex]x^2 = -1[/latex], the definition of [latex]b^{\frac{m}{n}}[/latex] when [latex]b[/latex] is negative and [latex]n[/latex] is even must involve the imaginary number [latex]i[/latex].

The following are rules for operations on numbers with rational exponents.

Multiplying Numbers with Rational Exponents

The following holds true for any rational exponent:

[latex]\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}[/latex]

For example, we can rewrite [latex]\sqrt[3]{16}[/latex] as a product:

[latex]\sqrt[3]{16}= \sqrt[3]{8} \cdot \sqrt[3]{2}[/latex]

Notice that [latex]\sqrt[3]{8} = 2[/latex], and therefore we have:

[latex]\sqrt[3]{16} = 2\sqrt[3]{2}[/latex]

Dividing Numbers with Rational Exponents

The following holds true for any rational exponent:

[latex]\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}[/latex]

For example, we can rewrite [latex]\sqrt{\frac{13}{9}}[/latex] as a fraction with two radicals:

[latex]\sqrt{\dfrac{13}{9}} = \dfrac{\sqrt{13}}{\sqrt{9}}[/latex]

Notice that the denominator can be simplified further:

[latex]\sqrt{9} = 3[/latex]

Therefore, the simplified form is:

[latex]\sqrt{\dfrac{13}{9}} = \dfrac{\sqrt{13}}{3}[/latex]

Canceling Powers and Roots

In some cases, writing an exponent in its fraction form makes it easier to cancel powers and roots. Recall that [latex]\sqrt[n]{{b}^{m}}= {({b}^{m})}^{\frac{1}{n}}= {b}^{\frac {m}{n}} [/latex]. We can use this rule to easily simplify a number that has both an exponent and a root.

For example, consider [latex]\sqrt[4]{5^8}[/latex]. This would take a long time to work out by hand, but consider how it can be rewritten using a rational exponent:

[latex]\sqrt[4]{5^8} = 5^{\frac{8}{4}}[/latex]

We can simplify the fraction in the exponent to 2, giving us [latex]5^2=25[/latex].

Example 1

Simplify the following expression:

[latex]\sqrt{ \dfrac{3^8}{25}}[/latex]

This expression can be rewritten using the rule for dividing numbers with rational exponents:

[latex]\sqrt{ \dfrac{3^8}{25}} = \dfrac{\sqrt{3^8}}{\sqrt{25}}[/latex]

Notice that the radical in the denominator is a perfect square and can therefore be rewritten as follows:

[latex]\sqrt{25} = 5[/latex].

Now, notice that the numerator can be rewritten:

[latex]\sqrt{3^8} = (3^8)^{\frac{1}{2}} = 3^{\frac{8}{2}}= 3^4[/latex].

Therefore, the simplified form is:

[latex]\sqrt{ \dfrac{3^8}{5}} = \dfrac{3^4}{5}[/latex]

Example 2

Simplify the following expression:

[latex]\dfrac{\sqrt{7^5}}{\sqrt[4]{7^2}}[/latex]

First, rewrite the numerator and denominator in rational exponent form:

[latex]\dfrac{7^{\frac{5}{2}}}{7^{\frac{2}{4}}}[/latex]

Notice that the exponent in the denominator can be simplified:

[latex]\dfrac{7^{\frac{5}{2}}}{7^{\frac{1}{2}}}[/latex]

Recall the rule for dividing numbers with exponents, in which the exponents are subtracted. Applying the division rule, we have:

[latex]\dfrac{7^{\frac{5}{2}}}{7^{\frac{1}{2}}} = 7^{\left(\frac{5}{2}-\frac{1}{2}\right)} = 7^{\frac{4}{2}} = 7^2[/latex]

Thus, the simplified form is simply [latex]7^2 = 49[/latex].

Scientific Notation

Scientific notation is used to express a very large or small number in the form [latex]m \cdot10^n[/latex], where [latex]m[/latex] has only one digit left of the decimal point.

Learning Objectives

Explain why scientific notation is useful in performing calculations with large or small numbers

Key Takeaways

Key Points

  • Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form.
  • In normalized scientific notation, the exponent [latex]n[/latex] is chosen so that the absolute value of [latex]m[/latex] remains at least 1 but less than 10 [latex]\left ( 1 \leq \left | m \right | < 10 \right )[/latex]—i. e. so that [latex]m[/latex] has exactly one digit left of its decimal point.
  • When numbers written in scientific notation are involved in multiplication or division, the standard rules for operations with exponentiation apply. When addition or subtraction is involved, the numbers must first be rewritten so the exponents are the same.
  • Most calculators present very large and very small results in scientific notation. Because superscripted exponents like [latex]10^7[/latex] cannot always be conveniently displayed, the letter E or e is often used to represent “times ten raised to the power of” (which would be written as “[latex]\cdot 10^b[/latex]“).

Key Terms

  • scientific notation: A method of writing or of displaying real numbers as a decimal number between 1 and 10 multiplied by an integer power of 10.
  • normalized scientific notation: A number written in scientific notation [latex]m \cdot 10^n[/latex] such that the absolute value of [latex]m[/latex] remains at least 1 but less than 10.

Scientific notation, also known as “standard form,” is a way to more conveniently write numbers that are very large or very small. This method is commonly used by mathematicians, scientists, and engineers.

For example, the numbers   [latex]43,000,000,000,000,000,000[/latex] (the number of different possible configurations of a Rubik’s cube) and [latex]0.000000000000000000000340[/latex] (the mass of the amino acid tryptophan) are extremely inconvenient to write and read. Therefore, they can be rewritten as a power of 10 using scientific notation.

Scientific notation is written as follows:

[latex]m \cdot 10^n[/latex]

This is read “[latex]m[/latex] times 10 raised to the power of [latex]n[/latex].”

How to Use Scientific Notation

To write a number in scientific notation:

  • Move the decimal point so that there is one nonzero digit to its left.
  • Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

For example, let’s write the number 43,500 in scientific notation. There are four digits in this number, so the decimal should be moved 4 places to the left to leave one nonzero digit left of the decimal point:

[latex]43,500 = 4.35 \cdot 10^{4}[/latex]

The exponent is -4 because the decimal point was moved to the left (the exponent would be positive had the decimal been moved to the right) by exactly 4 places.A number written in scientific notation can also be converted to standard form by reversing the process described above. For example, let’s write the number [latex]2.15 \cdot 10^{-3}[/latex] in standard form:

[latex]2.15 \cdot 10^{-3}= 0.00215[/latex]

To reverse the process, we move the decimal point three places to the left, adding leading zeroes where necessary.

Normalized Scientific Notation

Any given number can be written in the form of [latex]m \cdot 10^{n}[/latex] in many ways; for example, 350 can be written as [latex]3.5 \cdot 10^{2}[/latex] or [latex]35 \cdot 10^{1}[/latex]or [latex]350 \cdot 10^{0}[/latex], etc.

In normalized scientific notation, also called exponential notation, the exponent [latex]n[/latex] is chosen so that the absolute value of [latex]m[/latex] remains at least 1 but less than 10. In other words, [latex] 1 \leq | m | < 10 [/latex]. This form allows easy comparison of two numbers of the same sign with [latex]m[/latex] as a base, as the exponent [latex]n[/latex] gives the number’s order of magnitude.

Following these rules, 350 would always be written as [latex]3.5 \cdot 10^{2}[/latex] and [latex]0.015[/latex] would always be written as [latex]1.5 \cdot 10^{-2}[/latex].  Note that [latex]0[/latex] cannot be written in normalized scientific notation since it cannot be expressed as [latex]m \cdot 10^n[/latex] or any non-zero [latex]m[/latex].

Normalized scientific form is the typical form of expression for large numbers in many fields, except during intermediate calculations or when an unnormalized form, such as engineering notation, is desired.

Calculations involving Scientific Notation

When numbers written in scientific notation are multiplied or divided, the standard rules for operations with exponentiation apply. For example:

[latex]\begin{align} \displaystyle (3.12 \cdot 10^2) \cdot (4.06 \cdot 10^5) &= 3.12 \cdot 4.06 \cdot 10^{\left(2+5\right)} \\&=12.67 \cdot 10^7 \\&=1.267 \cdot 10^8 \end{align} [/latex]

[latex]\begin{align} \displaystyle \frac{1.85 \cdot 10^3}{4.25 \cdot 10^{-2}} &= \frac{1.85}{4.25} \cdot 10^{3-(-2)} \\&=.435 \cdot 10^5 \\&= 4.35 \cdot 10^4 \end{align}[/latex]

When numbers written in scientific notation are added to or subtracted from each other, the terms first must be rewritten so the exponents are the same. Then, the constant value, or [latex]m[/latex], can simply be added or subtracted. For example:

[latex]\begin{align} \displaystyle (3.12 \cdot 10^6)+(1.24 \cdot 10^7)&= (3.12 \cdot 10^6)+(12.4 \cdot 10^6) \\&= 15.52 \cdot 10^6 \end{align}[/latex]

E Notation

Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like [latex]10^7[/latex] cannot always be conveniently displayed, the letter E or e is often used to represent the phrase “times ten raised to the power of” (which would be written as “[latex]\cdot 10^n[/latex]“) and is followed by the value of the exponent. Note that in this usage, the character e is not related to the mathematical constant [latex]\mathbf{e}[/latex] or the exponential function [latex]e^x[/latex] (a confusion that is less likely if a capital E is used), and though it stands for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation. The use of this notation is not encouraged by publications, however.