Advanced Algebra 2 - Chapter 6 Review

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Questions and Answers

Rewrite 31/2 in radical form.

√3

Rewrite √9 in exponential form.

91/2

Rewrite (√10)³ in exponential form.

103/2

Evaluate √64 without using a calculator.

<p>8</p> Signup and view all the answers

Simplify the expression (2x1/4)(3x3/4y)(4y-1/3). Make sure all answers have positive exponents.

<p>24x²y1/3</p> Signup and view all the answers

Simplify the expression x1/2 / x2/3. Make sure all answers have positive exponents.

<p>x-1/6</p> Signup and view all the answers

Simplify the expression (-27x4y5)-1/3. Make sure all answers have positive exponents.

<p>1 / ( -3x4/3y5/3)</p> Signup and view all the answers

Simplify the expression x-1/2y1/3 / x1/3y2/3. Make sure all answers have positive exponents.

<p>1 / ( x5/6y1/3)</p> Signup and view all the answers

Simplify the expression √6(√7 – 3√2).

<p>√42 - 6√3</p> Signup and view all the answers

Simplify the expression (3 - √6) / (2 - √6).

<p>-√6 - 3</p> Signup and view all the answers

Simplify the expression (5 + √3)(6 - √5).

<p>33 + 6√3 - 5√5 - √15</p> Signup and view all the answers

Simplify the given radical. 3√320

<p>8√10</p> Signup and view all the answers

Simplify the given radical. 3√4 * 4√8

<p>48</p> Signup and view all the answers

Simplify the given radical. 81x³y⁸z⁶

<p>3xyz²√9xy⁶z⁴</p> Signup and view all the answers

Simplify the given radical. √24x³y⁹

<p>2xy⁴√6x</p> Signup and view all the answers

Simplify the given radical. 4√3 / √8. Be sure to rationalize the denominator.

<p>√6</p> Signup and view all the answers

Simplify the given radical. 3√2ab³c⁵ / 3√a²b⁵. Be sure to rationalize the denominator.

<p>c√6ac² / ab</p> Signup and view all the answers

Simplify the following radical expression. 2√3 + 9√3 + 3√4

<p>11√3 + 6</p> Signup and view all the answers

Simplify the following radical expression. 3√5 - 3√135

<p>-6√5</p> Signup and view all the answers

Simplify the following radical expression. 2√9x²y⁵ + 5xy√16y³.

<p>6xy²√y + 20xy²√y</p> Signup and view all the answers

Simplify the following radical expression. 4a³√b⁷ – 7b³√a³b⁴

<p>4a³b³√b – 7a³b³√b = -3a³b³√b</p> Signup and view all the answers

If f(x) = 2x + 1 and g(x) = 7x - 9, find (f + g)(x).

<p>9x - 8</p> Signup and view all the answers

If f(x) = 3x² and g(x) = -12x³, find (fg)(x).

<p>-36x⁵</p> Signup and view all the answers

If f(x) = x + 1 and g(x) = x² + 3x - 4, find f(g(3)).

<p>15</p> Signup and view all the answers

Solve the equation 2x⁵ + 36 = 100. Check for extraneous solutions. Write answers in simplest radical form.

<p>x = √2</p> Signup and view all the answers

Solve the equation 2(x - 3)⁴ - 12 = 50. Check for extraneous solutions. Write answers in simplest radical form.

<p>x = 3 ± √5</p> Signup and view all the answers

Solve the equation -4√x + 2 - 1 = 7. Check for extraneous solutions. Write answers in simplest radical form.

<p>x = 9</p> Signup and view all the answers

Solve the equation √7x - 4 - 4 = -6 . Check for extraneous solutions. Write answers in simplest radical form.

<p>x = 0</p> Signup and view all the answers

Solve the equation (8x)⁴/³ + 44 = 300. Check for extraneous solutions. Write answers in simplest radical form.

<p>x = 3</p> Signup and view all the answers

Solve the equation (x - 5)⁵/³ - 73 = 170. Check for extraneous solutions. Write answers in simplest radical form.

<p>x = 32</p> Signup and view all the answers

Solve the equation x +2 = √2x + 7. Check for extraneous solutions. Write answers in simplest radical form.

<p>x = 3</p> Signup and view all the answers

Find the inverse of the function f(x) = 2x -13.

<p>f⁻¹(x) = (x + 13)/2</p> Signup and view all the answers

The function graphed below has an inverse function.

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

Verify that the functions f(x) = 3x-4 and g(x) = (x – 4) / 3 are inverses.

<p>To show that two functions are inverses of each other, we need to show that f(g(x)) = x and g(f(x)) = x. f(g(x)) = f((x – 4) / 3) = 3((x – 4) / 3) – 4 = x – 4 – 4 = x – 8. This does not equal x, therefore the functions are not inverses.</p> Signup and view all the answers

Verify that the functions f(x) = √x - 4 + 1 and g(x) = (x - 1)² + 4 are inverses.

<p>To show that two functions are inverses of each other, we need to show that f(g(x)) = x and g(f(x)) = x. Let’s check f(g(x)) = f((x - 1)² + 4) = √((x-1)² + 4) - 4 + 1 = √(x² -2x+5) - 3. This is not equal to x. Therefore, f(x) and g(x) are not inverses.</p> Signup and view all the answers

Graph the function f(x) = 2√x + 1. Give the domain and range. Label your axes with an appropriate scale.

<p>Domain: x ≥ 0, Range: y ≥ 1. The graph is a vertical stretch of the graph y = √x by a factor of 2, followed by a vertical translation up 1 unit.</p> Signup and view all the answers

Let the graph of g be a vertical stretch by a factor of 2, followed by a translation 3 units left of the graph of f(x) = √x. Write a rule for g.

<p>g(x) = 2√(x + 3)</p> Signup and view all the answers

Let the graph of g be a horizontal shrink by a factor of ½, followed by a reflection in the x-axis and a vertical translation up 6 of the graph of f(x) = √x + 2. Write a rule for g.

<p>g(x) = -2√(2x) + 6</p> Signup and view all the answers

Describe the transformation of f(x) = √x represented by g(x) = -(1/3)√x - 3 + 1.

<p>This function is a vertical shrink by a factor of 1/3, a reflection across the x-axis, a horizontal translation 3 units to the right and a vertical translation 1 unit up.</p> Signup and view all the answers

Flashcards

Radical Form

Expression involving roots, like √x.

Exponent Form

A way to express numbers using powers, like x^n.

Square Root

A number that produces a specified quantity when multiplied by itself.

Positive Exponents

Exponents greater than zero, indicating repeated multiplication.

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Radical Simplification

The process of reducing a radical to its simplest form.

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Rational Denominator

Removing any radicals from the denominator in a fraction.

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Combining Radicals

Adding or subtracting radicals with the same index.

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Domain

The set of all possible input values (x) for a function.

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Range

The set of all possible output values (y) of a function.

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Function Composition

Combining functions, like f(g(x)).

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Extraneous Solution

A solution that emerges from the solving process but is not valid.

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Inverse Function

A function that reverses another function, like undoing it.

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Transformations

Changes to a graph, like shifts, stretches, or reflections.

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Quadratic Function

A polynomial function of degree 2, like f(x) = ax^2 + bx + c.

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Evaluate

To calculate the value of an expression.

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Polynomial

An algebraic expression with multiple terms, combined using addition and subtraction.

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Like Terms

Terms that have the same variable raised to the same power.

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Radical Expressions

Expressions that include a root, such as square roots.

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Factor

To break down an expression into smaller parts that multiply to the full expression.

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Graphing

Representing functions visually on a coordinate system.

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Radical Index

The root number in a radical, indicating the degree of the root.

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Simplifying Expressions

The process of rewriting an expression in a simpler form.

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Roots

Values of x that satisfy the equation where f(x) = 0.

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Solve Equations

Finding the values of variables that make the equation true.

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Coefficient

A number multiplying a variable in a term.

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Algebraic Expression

A mathematical phrase that can include numbers, variables, and operation signs.

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Study Notes

Advanced Algebra 2 - Chapter 6 Review

  • Review of radical form and exponents: Exercises include converting expressions between radical and exponent form (e.g., 31/2, x3/5).
  • Evaluating expressions without a calculator: Problems involve simplifying expressions containing radicals and fractional exponents (e.g., √64, 253/2).
  • Simplifying expressions with positive exponents: Focuses on simplifying expressions involving exponents and variables, ensuring all exponents are positive (e.g., (2x1/4)(3x3/4y)(4y-1/3)). Includes examples of negative exponents.
  • Simplifying radicals including rationalizing denominators: Exercises involve simplifying radicals and rationalizing denominators containing radicals (e.g., √6(√7 – 3√2), (3-√6) / (2-√6)).
  • Simplifying given radicals: Include exercises in simplifying cube roots and more complex radicals (e.g., 3√486, √54a8).
  • Simplifying radical expressions: Includes problems like combining like radicals (e.g., 2√3 + 9√3 + 3√4).
  • Function operations: Problems involve finding sums, differences, products, and quotients of functions (e.g., f(x) = 2x+1, g(x) = 7x-9; find f+g(x), f.g(x)).

Solving Equations and Checking for Extraneous Solutions

  • Equations involving radicals and exponents: Problems require solving equations with radicals (e.g., 2x5 + 36 = 100, 2(x – 3)4 – 12 = 50). Includes checking for extraneous solutions.
  • Solving radical equations: Examples involving square roots and other radicals in different forms (e.g., -4√x + 2x – 1 = 7).

Inverse Functions and Transformations

  • Finding inverse functions: Problems involve finding the inverse of given functions (e.g., f(x) = 2x–13, g(x) = 2x³-3).
  • Determining if a function has an inverse: Exercises involve graphical analysis determining if a graph represents a function that has an inverse.
  • Verifying that functions are inverses: Exercises involve showing that two given functions are inverses (f(x) = 3x - 4, g(x) = (x - 4) / 3).
  • Transformations of functions: Includes vertical stretches and shrinks, horizontal shifts and stretches, reflections, and vertical translations (e.g., graph y = 2√x + 1).

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