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Questions and Answers
Which of the following equals $3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$?
Which of the following equals $3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$?
- $3 \times 7$
- $3 + 7$
- $3^7$ (correct)
- $7^3$
Which of the following equals $(-9)^2$?
Which of the following equals $(-9)^2$?
- -18
- 81 (correct)
- 18
- -81
If $7^n \times a^m = a \times 7 \times a \times a \times a \times 7$, what is the value of n + m?
If $7^n \times a^m = a \times 7 \times a \times a \times a \times 7$, what is the value of n + m?
- 5 (correct)
- 3
- 6
- 2
Which of the following equals $2^{-4}$?
Which of the following equals $2^{-4}$?
Which of the following is the multiplicative inverse of the number $(-1)^3$?
Which of the following is the multiplicative inverse of the number $(-1)^3$?
Which of the following is the additive inverse of the number $4^{-3}$?
Which of the following is the additive inverse of the number $4^{-3}$?
Which of the following equals $a^{-1} \times a^3$?
Which of the following equals $a^{-1} \times a^3$?
Which of the following expresses $\frac{y^{-2}}{y^6}$ in its simplest form?
Which of the following expresses $\frac{y^{-2}}{y^6}$ in its simplest form?
$\frac{p^{-1}}{p^{-11}} = p^{10}$
$\frac{p^{-1}}{p^{-11}} = p^{10}$
$5_a^0 - (5_a)^0 = $
$5_a^0 - (5_a)^0 = $
If $X = \frac{1}{2}, Y = 3$ , what is the value of $XY$?
If $X = \frac{1}{2}, Y = 3$ , what is the value of $XY$?
If $2^4 \times a = 2^{20}$, what is the value of a?
If $2^4 \times a = 2^{20}$, what is the value of a?
If $2^{-5} \times a = 1$, what is the value of a?
If $2^{-5} \times a = 1$, what is the value of a?
$( \frac{1}{4} )^0 + \frac{1}{4} = $
$( \frac{1}{4} )^0 + \frac{1}{4} = $
$3^2 \times 2^2 = $
$3^2 \times 2^2 = $
Which of the following equals a third of the number $3^X$?
Which of the following equals a third of the number $3^X$?
Which of the following equals one quarter of the number $2^{20}$?
Which of the following equals one quarter of the number $2^{20}$?
Which of the following equals $2^A + 2^A$?
Which of the following equals $2^A + 2^A$?
Which of the following is equal to $\frac{1-3a^{-2}}{a^{-2}}$ in its simplest form?
Which of the following is equal to $\frac{1-3a^{-2}}{a^{-2}}$ in its simplest form?
Which of the following numbers is written in scientific notation?
Which of the following numbers is written in scientific notation?
Which of the following numbers is not in scientific notation?
Which of the following numbers is not in scientific notation?
Which of the following expresses the number 8 million in scientific notation?
Which of the following expresses the number 8 million in scientific notation?
Which of the following equals 0.000073?
Which of the following equals 0.000073?
If $6.3 \times 10^n = .00063$, what is the value of n?
If $6.3 \times 10^n = .00063$, what is the value of n?
If the number $y \times 10^9$ is written in scientific notation, what could be the value of y?
If the number $y \times 10^9$ is written in scientific notation, what could be the value of y?
If $39 \times 10^{-8} = k \times 10^{-7}$, what is the value of k?
If $39 \times 10^{-8} = k \times 10^{-7}$, what is the value of k?
Which of the following is the largest?
Which of the following is the largest?
Which of the following equals 6,000 \times 50?
Which of the following equals 6,000 \times 50?
Which of the following equals 0.7 \times 0.005?
Which of the following equals 0.7 \times 0.005?
If the speed of light is equal to 300,000 km/s what is speed of light in m/s?
If the speed of light is equal to 300,000 km/s what is speed of light in m/s?
Flashcards
Scientific Notation
Scientific Notation
A way to write very large or very small numbers using a product of a number between 1 and 10 and a power of 10.
Exponent
Exponent
The number of times a base is multiplied by itself.
Base
Base
The number being multiplied by itself when raised to a power.
Repeated Multiplication
Repeated Multiplication
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Power
Power
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Zero Exponent Rule
Zero Exponent Rule
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Square Root
Square Root
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Product of Powers
Product of Powers
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Quotient of Powers
Quotient of Powers
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Negative Exponent
Negative Exponent
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Study Notes
- Unit 1 is about Powers, Exponents, and Roots.
- Lesson One focuses on Powers and Exponents.
- Lesson Two is about Scientific Notation.
- Lesson Three covers topics on Square Roots and Cube Roots.
Learning Outcomes for Powers and Exponents
- Understanding the concept of repeated multiplication and its expression in exponential form is key.
- Distinguishing between the concepts of power and exponent is necessary.
- Applying the laws of exponents to solve problems in exercises is a learning outcome.
- Using laws of exponents to simplify mathematical expressions is an intended result.
- Using positive, negative, and zero exponents in solving exercises is a goal.
Vocabulary Related to Powers and Exponents
- Repeated Multiplication refers to the repeated product of the same factor.
- Exponent indicates the power to which a base is raised.
- Power is the result of raising a base to an exponent.
- Base is the number that is raised to a power.
Repeated Multiplication and Exponential Form
- Multiplying repeated factors can be shown with powers or in exponential form, with both exponent and base.
- Example: 3 x 3 x 3 x 3 can be expressed as 3 to the power of 4, which is written as exponent
Even and Odd Exponents of a Negative Base
- If 'a' is a positive number and 'm' is a positive integer, then (-a)^m = a^m when m is an even number.
- Example: (-3)^2 = 3^2 = 9
- If 'a' is a positive number and 'm' is a positive integer, then (-a)^m = -a^m when m is an odd number.
- Example: (-3)^3 = -3^3 = -27
- (-3)^2 = 9 but -3^2 = -9 is a distinction to consider.
Multiplication and Division of Powers with the Same Base
- When multiplying powers sharing a base, the base is kept, and the exponents are added.
- For a rational number 'a' and integers 'm, n': a^m x a^n = a^(m+n). meaning 7^8 x 7^4 x 7 = 7^(8+4+1) = 7^13
- When dividing powers with the same base, the base remains the same, and the exponents are subtracted.
- For any non-zero 'a', and integers 'm, n': a^m / a^n = a^(m-n). Meaning 8^6 / 8^2 = 8^(6-2) = 8^4.
Zero Exponent and Negative Integer Exponents
- Any number, not equal to zero, when raised to the power of zero, equals 1 with a^0 = 1 where a ≠0.
- Example: 7^0 = 1, (1/4)^0 = 1
- Any number, not equal to zero, raised to the power of (-n) is equal to the multiplicative inverse of the same number raised to the power of n with a^(-n) = 1/a^n where a ≠0.
- Example: 3^(-2) = 1/3^2 = 1/9, (1/7)^(-1) = (7/1)^1 = 7, (1/5)^(-2) = 5^2 = 25
Learning outcomes for Scientific Notation
- Expressing numbers using scientific notation is the objective.
- Converting between standard and scientific notation is a key ability.
- Comparing and ordering sets of numbers in scientific notation is imperative.
- Performing arithmetic operations with numbers in scientific notation is the purpose.
Vocabulary for Scientific Notation
- Scientific Notation - way of expressing numbers as a product of a number between 1 and 10 and a power of 10.
- Standard Form - usual way of writing numbers, without using powers of 10.
Understanding Scientific Notation
- Scientific notation writes very large or small numbers with two factors.
- One factor has an absolute value between 1 and 10.
- The other factor is 10 raised to an integer power.
- It is expressed as a x 10^n where: 1 ≤ |a| < 10 and n is an integer.
Writing Numbers in Scientific Notation
- For large numbers, move the decimal point to the left making the exponent of 10 positive.
- Example: 7,800,000 = 7.8 x 10^7
- For small numbers, move the decimal to the right making the exponent of 10 negative.
- Example: 0.000078 = 7.8 x 10^-5
Converting Numbers from Scientific Notation to Standard Form
- When 'n' is positive, move the decimal point 'n' places to the right.
- When 'n' is negative, move the decimal point |n| places to the left.
- Example: 3.7 x 10^5 = 370,000, writing the number in standard form
- Example: 7.38 x 10^-3 = 0.00738
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