Algebra 2 6-1 Guided Practice: Roots and Radical Expressions

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Solve It!
This equation contains an infinite radical.
x=\sqrt{1+\sqrt{1+\sqrt{1+...}}}​​​

Square each side. You get a quadratic equation that also contains an infinite radical.
x^2=1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}​​​

Are the two solutions of the quadratic equation also solutions of this equation? Explain.
Hint: consider substitution from one equation into the other.

Solution
No. Since x=\sqrt{1+\sqrt{1+\sqrt{1+...}}}, we can substitute x
into the right side of the equation and get x^{2}=1+x. Therefore, the solutions of the quadratic equation are both positive and negative, as follows: \frac{1\pm\sqrt{5}}{2}. The solution to the original equation is only positive.

Take Note: Take a moment to record the properties of exponents in your notes.

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Take Note: Use the properties of exponents to match equivalent exponential expressions below.

Draggable item		Corresponding Item
x^0 (x\neq0)		1
(xy)^3		\frac{1}{x^3}
x^{2}\cdot x^{1}		x^2+1
(x^{2})^3		x^5-3
{(\frac{x}{y})}^2		x^6
x^{-3}		\frac{x^2}{y^2}
\frac{x^{5}}{x^3}		x^{3}y^{3}

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Take Note: Use the TAKE NOTE Key Concept: The nth Root to identify the values of n and b that produce each scenario described on the right.

Recall: If a^{n}=b, with a and b real numbers and n a positive integer, then a is an nth root of b.

HINT: Each category on the right should contain two items from the left.

n is even
n is odd
b may be positive or negative
b is positive
b is negative

There is one real nth root of b, denoted in radical form as \sqrt[n]{b}.
There are two real nth roots of b, denoted in radical forms as \sqrt[n]{b} and -\sqrt[n]{b}.
There are no real nth roots of b.

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Take Note: Create a simplified radical expression with index 3 and radicand 7.

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Problem 1 Got It? What are the fifth roots of 0, -1, and 32?

-32
-2
-1
no real root
0
1
2
32

Fifth root of 0
Fifth root of -1
Fifth root of 32

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Problem 1 Got It? What are the real square roots of the following numbers?

-0.1
-\frac{36}{121}
-\frac{6}{11}
no real root
0
0.1
6/11
36/121

Real square root(s) of
Real square root(s) of
Real square root(s) of

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Problem 1 Got It? Reasoning: Explain why a negative real number b has no real nth roots if n is even.

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Take Note: Summarize the Fundamental Theorem of Algebra.

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Problem 2 Got It?

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Problem 2 Got It?

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Problem 2 Got It?

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Problem 2 Got It?

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Take Note: When is it necessary to use absolute value when evaluating the nth root of an ?

Some examples to consider:
\sqrt[3]{a^{3}}
\sqrt[4]{a^{4}}

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Problem 3 Got It?

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Problem 3 Got It?

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Algebra 2 6-1 Guided Practice: Roots and Radical Expressions

Video Check: Select all that apply with regards to the video embedded directly above this item.

Solve It!

Solution

Video Check: Select all that apply with regards to the video embedded directly above this item.

Take Note: Use the properties of exponents to match equivalent exponential expressions below.

Take Note: Create a simplified radical expression with index 3 and radicand 7.

Problem 1 Got It? What are the fifth roots of 0, -1, and 32?

Problem 1 Got It? What are the real square roots of the following numbers?

Problem 1 Got It? Reasoning: Explain why a negative real number b has no real nth roots if n is even.

Video Check: Select all that apply with regards to the video embedded directly above this item.

Take Note: Summarize the Fundamental Theorem of Algebra.

Problem 2 Got It?

Problem 2 Got It?

Problem 2 Got It?

Problem 2 Got It?

Video Check: Select all that apply with regards to the video embedded directly above this item.

Take Note: When is it necessary to use absolute value when evaluating the nth root of an ?Some examples to consider:\sqrt[3]{a^{3}} \sqrt[4]{a^{4}}

Problem 3 Got It?

Problem 3 Got It?

Video Check: Select all that apply with regards to the video embedded directly above this item.

🧠 Retrieval Practice:Summarize the mathematical content of this lesson. What topics, ideas, and vocabulary were introduced?

Video Check: Select all that apply with regards to the video embedded directly above this item.

Solve It!

Solution

Video Check: Select all that apply with regards to the video embedded directly above this item.

Take Note: Use the properties of exponents to match equivalent exponential expressions below.

Take Note: Create a simplified radical expression with index 3 and radicand 7.

Problem 1 Got It? What are the fifth roots of 0, -1, and 32?

Problem 1 Got It? What are the real square roots of the following numbers?

Problem 1 Got It? Reasoning: Explain why a negative real number b has no real nth roots if n is even.

Video Check: Select all that apply with regards to the video embedded directly above this item.

Take Note: Summarize the Fundamental Theorem of Algebra.

Problem 2 Got It?

Problem 2 Got It?

Problem 2 Got It?

Problem 2 Got It?

Video Check: Select all that apply with regards to the video embedded directly above this item.

Take Note: When is it necessary to use absolute value when evaluating the nth root of an ?Some examples to consider:\sqrt[3]{a^{3}} \sqrt[4]{a^{4}}

Problem 3 Got It?

Problem 3 Got It?

Problem 3 Got It? Hint: Keep in mind that x^{12} is never negative.x^{3}, on the other hand, can be negative. Use absolute value to ensure that the simplified expression remains equivalent to the original for all values of x.

Video Check: Select all that apply with regards to the video embedded directly above this item.

Take Note: It is important to recognize expanded forms of radicands when simplifying radical expressions.Which expressions are equivalent to \sqrt{25x^{8}}?Select all that apply.

Problem 4 Got It? Academics: Some teachers adjust test scores when a test is difficult. One teacher's formula for adjusting scores is show below, where A is the adjusted score and R is the raw score. What are the adjusted scores for raw scores of 0 and 100?

🧠 Retrieval Practice:Summarize the mathematical content of this lesson. What topics, ideas, and vocabulary were introduced?

Problem 3 Got It? Hint: Keep in mind that x^{12} is never negative.x^{3}, on the other hand, can be negative. Use absolute value to ensure that the simplified expression remains equivalent to the original for all values of x.

Take Note: It is important to recognize expanded forms of radicands when simplifying radical expressions.Which expressions are equivalent to \sqrt{25x^{8}}?Select all that apply.

Problem 4 Got It? Academics: Some teachers adjust test scores when a test is difficult. One teacher's formula for adjusting scores is show below, where A is the adjusted score and R is the raw score. What are the adjusted scores for raw scores of 0 and 100?

Take Note: When is it necessary to use absolute value when evaluating the nth root of an ?

Some examples to consider:
\sqrt[3]{a^{3}}
\sqrt[4]{a^{4}}

🧠 Retrieval Practice:
Summarize the mathematical content of this lesson. What topics, ideas, and vocabulary were introduced?

Take Note: When is it necessary to use absolute value when evaluating the nth root of an ?

Some examples to consider:
\sqrt[3]{a^{3}}
\sqrt[4]{a^{4}}

Problem 3 Got It?
Hint: Keep in mind that x^{12} is never negative.
x^{3}, on the other hand, can be negative. Use absolute value to ensure that the simplified expression remains equivalent to the original for all values of x.

Take Note: It is important to recognize expanded forms of radicands when simplifying radical expressions.
Which expressions are equivalent to \sqrt{25x^{8}}?
Select all that apply.

Problem 4 Got It? Academics: Some teachers adjust test scores when a test is difficult. One teacher's formula for adjusting scores is show below, where A is the adjusted score and R is the raw score.
What are the adjusted scores for raw scores of 0 and 100?

🧠 Retrieval Practice:
Summarize the mathematical content of this lesson. What topics, ideas, and vocabulary were introduced?

Problem 3 Got It?
Hint: Keep in mind that x^{12} is never negative.
x^{3}, on the other hand, can be negative. Use absolute value to ensure that the simplified expression remains equivalent to the original for all values of x.

Take Note: It is important to recognize expanded forms of radicands when simplifying radical expressions.
Which expressions are equivalent to \sqrt{25x^{8}}?
Select all that apply.

Problem 4 Got It? Academics: Some teachers adjust test scores when a test is difficult. One teacher's formula for adjusting scores is show below, where A is the adjusted score and R is the raw score.
What are the adjusted scores for raw scores of 0 and 100?