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Questions and Answers
Simplify the expression: (x^3/2 * x^1/4)^2
Simplify the expression: (x^3/2 * x^1/4)^2
If the nth term of a geometric sequence is given by an = 3 * (1/2)^(n-1), what is the common ratio?
If the nth term of a geometric sequence is given by an = 3 * (1/2)^(n-1), what is the common ratio?
Simplify the expression: √(27x^6y^3)
Simplify the expression: √(27x^6y^3)
Which of the following is equivalent to (1/2)^-3 ?
Which of the following is equivalent to (1/2)^-3 ?
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What is the value of x in the equation 2^x = 16?
What is the value of x in the equation 2^x = 16?
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Simplify the expression: √(18x^2y^5) / √(2xy)
Simplify the expression: √(18x^2y^5) / √(2xy)
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The graph of the exponential function f(x) = 2^x is shifted 3 units to the right and 2 units down. What is the equation of the transformed function?
The graph of the exponential function f(x) = 2^x is shifted 3 units to the right and 2 units down. What is the equation of the transformed function?
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What is the simplified form of the expression (x^-2 * y^3)^-1?
What is the simplified form of the expression (x^-2 * y^3)^-1?
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The first term of a geometric sequence is 5 and the common ratio is 2. What is the 5th term of the sequence?
The first term of a geometric sequence is 5 and the common ratio is 2. What is the 5th term of the sequence?
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Simplify the radical: √(48x^7y^9)
Simplify the radical: √(48x^7y^9)
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Flashcards
Exponents
Exponents
Represent repeated multiplication of a base by itself.
Exponential Functions
Exponential Functions
Functions in the form f(x) = ax, where 'a' is a positive constant and 'x' is a real number.
Rational Exponents
Rational Exponents
Exponents that are fractions, representing roots and powers.
Product Rule of Exponents
Product Rule of Exponents
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Quotient Rule of Exponents
Quotient Rule of Exponents
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Zero Exponent Rule
Zero Exponent Rule
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Negative Exponents
Negative Exponents
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Radical Expressions
Radical Expressions
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Common Ratio
Common Ratio
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Geometric Sequence Formula
Geometric Sequence Formula
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Study Notes
Exponents and Exponential Functions
- Exponents represent repeated multiplication. A base raised to an exponent indicates how many times the base is multiplied by itself.
- Exponential functions have the form f(x) = ax, where 'a' is a positive constant (the base) and 'x' is any real number.
- Exponential functions exhibit rapid growth or decay depending on the value of 'a' (the base). If 'a' is greater than 1, the function grows exponentially. If 'a' is between 0 and 1, the function decays exponentially.
Rational Exponents and Properties of Exponents
- Rational exponents are exponents that are fractions. A rational exponent 'm/n' represents the nth root of the base raised to the mth power (√nbm). This can be written as bm/n.
- Properties of exponents apply to rational exponents, including the product rule (bm * bn = bm+n), the quotient rule (bm / bn = bm-n), and the power rule ( (bm)n=bmn).
- Zero exponent for any non-zero base results in 1 (b0 = 1).
- Negative exponents indicate reciprocal (b-n = 1/bn)
- These rules help simplify expressions involving exponents to their simplest forms.
- Understanding negative exponents allows manipulating expressions with negative exponents more easily.
Radical Expressions
- Radical expressions are expressions containing radicals (√).
- Radical expressions can be written using rational exponents as shown above.
- The root of a radical (such as the square root, cube root, etc.) is represented by the index of the radical.
- Simplifying radical expressions involves rewriting them in their simplest forms.
- When combining radical expressions, you can add or subtract like radicals only after simplifying them (radicals with the same index and radicand).
- Operations like multiplication and division often involve common rationalization techniques. For example:√(a/b) can be written as √a/ √b
Geometric Sequences
- Geometric sequences are sequences of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio.
- The general form of a geometric sequence is given by an = a1 * r(n-1), where:
- an is the nth term in the sequence
- a1 is the first term
- r is the common ratio
- n is the term number
- The common ratio 'r' can be found by dividing any term by the preceding term.
- Geometric Sequences can have terms that are increasing or decreasing (depending on common ratio).
- The sum of a finite geometric sequence can be calculated using appropriate formula.
- Infinite geometric series can have a finite sum, under particular circumstances with relation to the common ratio.
Translations of Exponential Functions
- Translations of exponential functions are shifts of the graph horizontally or vertically.
- Adding a constant to the exponent translates the graph horizontally.
- Adding a constant to the function translates the graph vertically.
- The horizontal translation is based on the opposite value of the added constant.
- Exponential function transformations follow similar principles to transformations of other types of functions.
- Understanding horizontal and vertical shifts is crucial for graphing and analyzing exponential functions.
Studying That Suits You
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Description
Test your understanding of exponents and exponential functions, including their definitions and properties. This quiz will challenge you on both standard and rational exponents. Prepare to explore the rapid growth and decay of exponential functions.
