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

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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Advanced Algebra 2 Chapter 6 Review PDF

Description

This quiz covers key concepts from Chapter 6 of Advanced Algebra 2, focusing on radical forms and exponents. Students will explore exercises involving conversions between radical and exponent forms, evaluating expressions without calculators, and simplifying expressions with positive exponents. Additional topics include rationalizing denominators and simplifying various radicals.

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