Ch 6: Exponents Flashcards
Introducing Exponents: Positive Exponents
The expression 2 * 2 * 2 can be written using a base and an exponent.
2 * 2 * 2 = 2³
The factor 2 is the base. The exponent is 3 because the factor appears 3 times.
An expression such as 2³ is called a power. Read the expression 2³ as 2 to the third power or 2 raised to the third power.
An exponent may be positive, negative, or zero.
- A positive exponent indicates how many times the base is to be used as a factor.
- A negative exponent indicates how many times the reciprocal of the base is to be used as a factor.
- A zero exponent with a nonzero base indicates that the value of the power is 1.
- The expression of 0⁰ is undefined.
When no exponent is written in an expression, the exponent is understood to be 1.
Examples: 4 = 4¹, -7 = (-7)¹, 2/3 = (2/3)¹
Order of Operations
What is the answer to a problem like 3 + 5 * 6²? The order of operations that was introduced in Chapter 2 has to be expanded to take care of exponents. The new step is listed below.
- Complete all calculations within grouping symbols such as parentheses.
- -Evaluate all numbers with exponents in order from left to right.
- Perform all multiplications and divisions in order from left to right.
- Perform all additions and subtractions in order from left to right.
Applying Laws of Exponents: Multiplication
To find the area of the rectangle at the left, find the product 2⁴ * 2³ = (2 * 2 * 2 * 2) + (2 * 2 * 2) = 2⁷
To multiply powers that have the same base, add the exponents and use the same base.
2⁴ * 2³ = 2⁴⁺³ = 2⁷
Product of Powers
For any number except a = 0, and all numbers m and n: aⁿ * a(m = any power) = aⁿ⁺(m power)
Division
To divide powers that have the same base, subtract the exponents and use the same base. Always subtract the exponent in the denominator from the exponent in the numerator.
Quotient of Powers
For any number a except a = 0, and all numbers m and n: a(m power)/aⁿ = a(m power)⁻ⁿ
Power of a Quotient
A power of a quotient (fraction) can be evaluated by methods shown previously or by using the following law.
Power of a Quotient
For any nonzero numbers a and b, and all numbers n: (a/b)ⁿ = aⁿ/bⁿ
Power of a Power
An expression such as (a/b)ⁿ is known as a power of a power. To simplify a power of a power, multiply the exponents and use the same base.
Using Scientific Notation: Standard Form to Scientific Notation
Scientific notation is a useful way to represent very large or very small quantities.
A number in scientific notation is a product of the form: (a number that is a least 1 and less than 10) * (a power of 10)
To write a number in scientific notation, first move the decimal point to the immediate right of the first nonzero digit. Then count the number of places you moved the decimal point. Use that number as the exponent of 10. If the decimal point was moved to the left, the exponent is positive.
Write 35,000,000 in scientific notation
35,000,000 = 3.5 * 10⁷
Scientific Notation to Standard Form
A number in scientific notation can be written in standard form by moving the decimal point and filling in zeros as necessary. Use the exponent of 10 as the number of places to move the decimal point. If the exponent is positive, move the decimal point right; if the exponent is negative, move the decimal point left.
Write 2.0 * 10⁻⁵ in standard form
2.0 * 10⁻⁵ = 0.00002
Simplifying Expressions With Powers of Ten: Multiplication
To multiply two numbers that are powers of ten, add the exponent and use 10 as the base. This follows the rules for multiplication with exponents.
Example
10³ * 10⁴ = 10³⁺⁴ = 10⁷
Division
To divide two numbers that are powers of ten, subtract the exponents and use 10 as the base. This follows the rules for division with exponents.
—INSERT EXAMPLE HERE—[pg. 140]
Using Engineering Notation
Engineering notation is similar to scientific notation, but it requires that the exponent in the power of 10 be divisible by 3. The multiplier in front of the power of 10 must be at least 1 but less than 1,000.
To write a number in engineering notation, write it as a product in the following form:
(a number that is at least 1 and less than 1,000) * (a power of 10 whose exponent is divisible by 3)
Write 0.0351 in engineering notation
0.0351 = 35.1 * 10⁻³
“Move the decimal point 3 places to the right to form a number that is at least 1 and less than 1,000 multiplied by a power of 10 whose exponent is divisible by 3.”
Electronic multimeters, used to measure voltage, current, and resistance, are designed for used with engineering notation. A measurement of 4.05 * 10⁻⁶ amp could easily be found.
Introducing Roots and Fractional Exponents: Radicals
The square of a number is the second power of that number. The square of 6 is 36 because 6² = 6 * 6 = 36
A square root of a number is a number that can be squared to get the given number. The square roots of 36 are 6 and -6 because 6² = (6)(6) = 36 and (-6)² = (-6)(-6) = 36
Every positive number has a positive square root and a negative square root. We are normally concerned with the positive square root of a number, so we call it the principal square root. The principal square root of a number is indicated by the radical sign, √.
From now on, “square root” will mean “principal square root”.
Evaluate √8,100
90² = 8,100
So, the square root of 8,100 is 90
√8,100 = 90
There is an algorithm, or set of steps, that can be sued to find or approximate the square root of any positive number
Product Property of Square Roots
For any numbers a and b, where a > 0 and b > 0, √ab = √a * √b
Fractional Exponents
The square root of a number can be expressed using the fraction “1/2” as the exponent.
5(exponent of 1/2) * 5(exponent of 1/2) = 5
√5 * √5 = 5
So, √5 = 5(expnt of 1/2)
