Algebra Class: Exponents and Radicals
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Questions and Answers

What is the result of simplifying the expression 2√30 + 7√10?

  • 70√2
  • 20√15
  • 140√3 (correct)
  • 14√6

Which property allows the expression √ab to be expressed as √a * √b?

  • Exponent Law
  • Difference Property of Square Roots
  • Product Property of Square Roots (correct)
  • Sum Property of Square Roots

In the expression 2√30 + 7√10, which numbers are considered coefficients of the square roots?

  • 30 and 10
  • √30 and √10
  • 2 and 7 (correct)
  • 2√30 and 7√10

How can the radicand in the expression 2√(2 * 3 * 5) + 7√(2 * 5) be expressed without perfect square factors?

<p>2√2 * √3 * √5 + 7√2 * √5 (C)</p> Signup and view all the answers

What is the general form of an exponential function?

<p>y = ab^x (D)</p> Signup and view all the answers

In the exponential function represented by the table, what is the initial amount?

<p>3 (D)</p> Signup and view all the answers

What is the constant ratio of the exponential function based on the given data?

<p>4 (C)</p> Signup and view all the answers

Which of the following correctly represents the exponential function derived from the table?

<p>f(x) = 3 imes 4^x (C)</p> Signup and view all the answers

What condition must the constant ratio $b$ satisfy for it to be valid in an exponential function?

<p>b &gt; 0 and b ≠ 1 (D)</p> Signup and view all the answers

What happens to the graph of $g(x) = a^{x} + k$ when $k > 0$?

<p>The graph is translated up. (B)</p> Signup and view all the answers

How is the graph of $g(x) = a^{x-h}$ transformed when $h < 0$?

<p>The graph is translated horizontally left. (D)</p> Signup and view all the answers

If $f(x) = 3^{x}$, what is the value of $g(0)$ for $g(x) = 3^{x} - 2$?

<p>-1 (A)</p> Signup and view all the answers

Which of the following statements about the graphs of $f(x)$ and $g(x)$ is true?

<p>The graph of $g(x)$ is translated 2 units down from $f(x)$. (A)</p> Signup and view all the answers

What is the output of $f(-2)$ if $f(x) = 3^{x}$?

<p>1/9 (B)</p> Signup and view all the answers

What is the formula for an exponential growth function?

<p>$f(x) = a(1 + r)^x$ (B)</p> Signup and view all the answers

If a population of 18,000 grows at an annual rate of 8%, what will be the population after 1 year?

<p>19,440 (D)</p> Signup and view all the answers

In the function $f(x) = a(1 + r)^x$, what does 'r' represent?

<p>The growth rate (B)</p> Signup and view all the answers

What is the expected population of Chapter City after 6 years, given an initial population of 18,000 and an 8% growth rate?

<p>28,563.74 (B)</p> Signup and view all the answers

Which of the following best describes an exponential decay function?

<p>$f(x) = a(1 - r)^x$ (C)</p> Signup and view all the answers

What is the Power of a Power property used for?

<p>To simplify expressions with exponents (A)</p> Signup and view all the answers

When solving $64^{x-3}=16^{2x-1}$, what is the first step?

<p>Rewrite with the same base (B)</p> Signup and view all the answers

After applying the power of a power property to $64^{x-3}$ and $16^{2x-1}$, what do the exponents become?

<p>6(x-3) and 4(2x-1) (D)</p> Signup and view all the answers

What is the equation formed after equating the exponents from $64^{x-3}$ and $16^{2x-1}$?

<p>6x - 18 = 8x - 4 (D)</p> Signup and view all the answers

What is the common ratio of the geometric sequence 9, 22.5, 56.25, 140.625, 351.5625?

<p>2.5 (A)</p> Signup and view all the answers

What is the explicit formula for the geometric sequence starting with 9 and a common ratio of 2.5?

<p>$a_n = 9(2.5)^{n-1}$ (C)</p> Signup and view all the answers

What is the final value of x when solving for x in $64^{x-3}=16^{2x-1}$?

<p>-7 (A)</p> Signup and view all the answers

Which of the following is the correct recursive formula for the sequence?

<p>$a_n = 2.5(a_{n-1}), a_1 = 9$ (C)</p> Signup and view all the answers

Which term represents the fourth term in the given geometric sequence?

<p>140.625 (C)</p> Signup and view all the answers

What pattern can be observed in the geometric sequence provided?

<p>Each term is multiplied by 2.5 to get the next term. (C)</p> Signup and view all the answers

Flashcards

Exponential Function

A function where the independent variable (x) appears as an exponent of a constant base.

What is an exponential function?

A function where the independent variable (x) appears as an exponent.

Initial Amount (Exponential Function)

The value of the function when the independent variable (x) is zero.

Constant Ratio (Exponential Function)

The constant factor by which the function is multiplied for each one-unit increase in the independent variable (x).

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Coefficient 'a' in f(x) = a * b^x

A non-zero constant that determines the starting value of the function.

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Radical expression

A mathematical expression that involves a radical symbol, typically a square root.

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Product Property of Square Roots

The product of two or more square roots is the square root of the product of the radicands.

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Initial amount

The initial value or starting amount in an exponential function. It's the value when the exponent is zero.

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Constant ratio

The constant factor that is raised to a power in an exponential function. It represents the rate of change or growth.

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Vertical Translation of Exponential Functions (k)

A vertical shift of the graph of an exponential function, up when k > 0 and down when k < 0.

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Horizontal Translation of Exponential Functions (h)

A horizontal shift of the graph of an exponential function, right when h > 0 and left when h < 0.

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Vertical Translation Effect on the Graph

When comparing the graphs of $g(x)=a^{x}+k$ and $f(x)=a^{x}$, the graph of g(x) is shifted up or down by k units.

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Horizontal Translation Effect on the Graph

When comparing the graphs of $g(x) = a^{x-h}$ and $f(x)=a^{x}$, the graph of g(x) is shifted left or right by h units.

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Transformations of Exponential Functions

The graph of an exponential function is translated by adding a constant to the exponent (horizontal shift) or to the entire function (vertical shift).

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Geometric Sequence

A sequence where each term is found by multiplying the previous term by a constant value called the common ratio.

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Common Ratio (Geometric Sequence)

The constant factor by which each term in a geometric sequence is multiplied to get the next term.

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Explicit Formula (Geometric Sequence)

A formula describing the relationship between the nth term of a geometric sequence and the common ratio.

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Recursive Formula (Geometric Sequence)

A formula that describes the relationship between each term of a geometric sequence and the previous term.

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What is the Explicit Formula for a Geometric Sequence?

A formula that defines the nth term of a geometric sequence as a function of the common ratio (r) and the first term ($a_1$).

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Exponential Growth Function

A function that increases rapidly over time, with the growth rate proportional to the current value. It can be represented by the formula f(x) = a(1 + r)^x, where 'a' is the initial amount, 'r' is the growth rate, and 'x' is the time.

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Exponential Decay Function

A function that decreases rapidly over time, with the rate of decrease proportional to the current value. It can be represented by the formula f(x) = a(1 - r)^x where 'a' is the initial amount, 'r' is the decay or decrease rate, and 'x' is the time.

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Common Ratio

The constant factor by which each term in a geometric sequence is multiplied to get the next term. For example, in the sequence 2, 4, 8, 16 ..., the common ratio is 2.

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Term Number

The number of times the common ratio is multiplied to reach a specific term in a geometric sequence. For example, in the sequence 2, 4, 8, 16..., the term 16 is the 4th term.

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Rational Number

A number that can be expressed as a fraction, where the numerator and denominator are integers. Rational numbers can be written in decimal form, but the decimal representation will either terminate or repeat.

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Irrational Number

A number that cannot be expressed as a simple fraction (a/b) where a and b are integers. Irrational numbers are non-terminating and non-repeating decimals.

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Rational Exponent

An exponent that is a fraction. The numerator of the fraction represents the power to which the base is raised, and the denominator represents the root to be taken.

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Power of a Power Property

A property of exponents that states that when raising a power to another power, you multiply the exponents. For example, (a^m)^n = a^(m*n).

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Study Notes

Exponential Functions

  • An exponential function is the product of an initial amount and a constant ratio raised to a power.
  • Exponential functions are expressed using f(x) = a ⋅ bx, where a is a nonzero constant, b > 0, and b ≠ 1.
  • The initial amount is the value of the function when x = 0.
  • The constant ratio (common ratio) is the value by which the function is multiplied each time x increases by 1.
  • To find the constant ratio, divide any output value by the previous output value.
  • To find the initial amount, look for the output value when x = 0.

Example

  • Find the initial amount and the constant ratio of the exponential function represented by the table.
x f(x)
0 3
1 12
2 48
3 192
4 768
  • The initial amount is 3.
  • f(0) = 3
  • The constant ratio is 4.
  • 12 ÷ 3 = 4
  • 48 ÷ 12 = 4
  • 192 ÷ 48 = 4
  • 768 ÷ 192 = 4
  • In f(x) = a ⋅ bx, substitute 3 for a and 4 for b.
  • The function is f(x) = 3(4)x.

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Test your understanding of exponents and radicals with this quiz. Explore simplification of expressions, properties of square roots, and characteristics of exponential functions. It's a great way to reinforce your algebra skills.

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