Negative Exponents and Scientific Notation: Study Guide

Section 5.7: Negative Exponents and Scientific Notation

Negative Exponents

Negative exponents are a fundamental concept in algebra, allowing us to express reciprocals and simplify expressions. Understanding how to manipulate negative exponents is essential for working with exponential expressions and scientific notation.

• Negative Exponent Rule: If b is any real number other than 0 and n is a natural number, then: $b^{-n} = \frac{1}{b^{n}}$ $\frac{1}{b^{-n}} = b^{n}$

• Switching Positions: When a negative exponent appears, move the base from numerator to denominator (or vice versa) and make the exponent positive. The sign of the base itself does not change.

• Example: $x^{-3} = \frac{1}{x^{3}}$

• Example: $\frac{1}{y^{-2}} = y^{2}$

Writing Expressions with Positive Exponents

To simplify expressions, it is often necessary to rewrite all exponents as positive.

• Example: $a^{-4} = \frac{1}{a^{4}}$

• Example: $\frac{m^{-2}}{n^{-5}} = \frac{n^{5}}{m^{2}}$

• Example: $\left(\frac{x^{-1}}{y^{3}}\right) = \frac{1}{x y^{3}}$

Simplifying Exponential Expressions

An exponential expression is considered simplified when:

• Each base occurs only once.

• No parentheses appear.

• No powers are raised to powers.

• No negative or zero exponents appear.

To simplify, use the following properties:

• Product Rule: $a^{m} \cdot a^{n} = a^{m+n}$

• Quotient Rule: $\frac{a^{m}}{a^{n}} = a^{m-n}$

• Power Rule: $(a^{m})^{n} = a^{m \cdot n}$

• Negative Exponent Rule: $a^{-n} = \frac{1}{a^{n}}$

Example: Simplify $\frac{x^{3} y^{-2}}{x^{-1} y^{4}}$

• Apply quotient rule: $x^{3} / x^{-1} = x^{3 - (-1)} = x^{4}$

• Apply quotient rule: $y^{-2} / y^{4} = y^{-2 - 4} = y^{-6}$

• Rewrite negative exponent: $y^{-6} = \frac{1}{y^{6}}$

• Final answer: $\frac{x^{4}}{y^{6}}$

Scientific Notation

Scientific notation is a method for expressing very large or very small numbers in a compact form. A positive number is written in scientific notation as:

• $a \times 10^{n}$, where $1 \leq a < 10$ and $n$ is an integer.

• Example: $4,500,000 = 4.5 \times 10^{6}$

• Example: $0.00032 = 3.2 \times 10^{-4}$

Converting from Scientific Notation to Decimal Notation

To convert a number from scientific notation to decimal notation:

• If $n$ is positive, move the decimal point in $a$ to the right $n$ places.

• If $n$ is negative, move the decimal point in $a$ to the left $|n|$ places.

• Example: $7.1 \times 10^{9}$ becomes $7,100,000,000$

• Example: $2.5 \times 10^{-6}$ becomes $0.0000025$

Converting from Decimal Notation to Scientific Notation

To convert a decimal number to scientific notation:

• Move the decimal point to create a number $a$ such that $1 \leq a < 10$.

• Count the number of places moved; this is $|n|$.

• If the original number is greater than 10, $n$ is positive; if between 0 and 1, $n$ is negative.

• Example: $0.00057 = 5.7 \times 10^{-4}$

• Example: $23,000 = 2.3 \times 10^{4}$

Computations with Scientific Notation

Scientific notation simplifies calculations with very large or small numbers. The rules for operations are:

• Multiplication: Multiply the decimal parts and add the exponents. $(a \times 10^{m}) \times (b \times 10^{n}) = (a \times b) \times 10^{m+n}$

• Division: Divide the decimal parts and subtract the exponents. $\frac{a \times 10^{m}}{b \times 10^{n}} = \left(\frac{a}{b}\right) \times 10^{m-n}$

• Exponentiation: Raise the decimal part to the power and multiply the exponents. $(a \times 10^{m})^{k} = a^{k} \times 10^{m \cdot k}$

• After computation, adjust the result to ensure the decimal part is between 1 and 10.

Example: $(3.2 \times 10^{5}) \times (2 \times 10^{3}) = 6.4 \times 10^{8}$

Example: $\frac{4.5 \times 10^{7}}{1.5 \times 10^{2}} = 3 \times 10^{5}$

Applied Problems Using Scientific Notation

Scientific notation is useful in real-world applications, such as calculating prices, population, or scientific measurements.

• Example: The area of Alaska is approximately $1.7 \times 10^{9}$ acres. The purchase price was $7.2 \times 10^{6}$ dollars. To find the price per acre: $\frac{7.2 \times 10^{6}}{1.7 \times 10^{9}} = 4.24 \times 10^{-3}$ dollars/acre This equals about $0.00424$ per acre, or approximately 0.4 cents per acre.

• Example: If $787 \times 10^{9}$ dollars were divided among $3.08 \times 10^{8}$ people: $\frac{787 \times 10^{9}}{3.08 \times 10^{8}} = 2.55 \times 10^{3}$ dollars per person Each citizen would pay about $2,560$.

Summary Table: Scientific Notation Operations

Operation	Rule	Example
Multiplication	$(a \times 10^{m}) \times (b \times 10^{n}) = (a \times b) \times 10^{m+n}$	$(2 \times 10^{3}) \times (3 \times 10^{4}) = 6 \times 10^{7}$
Division	$\frac{a \times 10^{m}}{b \times 10^{n}} = \left(\frac{a}{b}\right) \times 10^{m-n}$	$\frac{8 \times 10^{6}}{2 \times 10^{2}} = 4 \times 10^{4}$
Exponentiation	$(a \times 10^{m})^{k} = a^{k} \times 10^{m \cdot k}$	$(5 \times 10^{2})^{3} = 125 \times 10^{6} = 1.25 \times 10^{8}$

Additional info: Examples and context were inferred and expanded for clarity and completeness.