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
{(\frac{x}{y})}^2		\frac{1}{x^3}
x^{2}\cdot x^{1}		x^2+1
\frac{x^{5}}{x^3}		x^5-3
(x^{2})^3		x^6
(xy)^3		\frac{x^2}{y^2}
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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Take Note: Explain why that is.

Keep in mind that the original expression and the simplified expression should be equivalent for all values of the variables, positive and negative.

It may also be helpful to consider that \sqrt{x^{2}}\neq x. That's because when x is negative, the expression on the left is positive and the expression on the right is negative.

On the other hand, adding an absolute value resolves that issue: \sqrt{x^{2}}=|x|.

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Problem 3 Got It?

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Problem 3 Got It?

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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.

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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.

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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?

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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?

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?