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Elementary Algebra
Roots and Radicals

1
Definitions and Common Roots

Problem  1

Find the following roots.

  1. \(\sqrt{36}\)

  2. \(-\sqrt{36}\)

  3. \(\sqrt{\dfrac{1}{36}}\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  2

Find the following roots, if possible

  1. \(\sqrt[3]{-8}\)

  2. \(\sqrt[3]{-64}\)

  3. \(\sqrt{-4}\)

  4. \(-\sqrt{4}\)

  5. \(\sqrt[4]{-16}\)

  6. \(-\sqrt[4]{16}\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  3
practice icon

Simplify \(\sqrt{25a^2}\), assume \(a\geq0\).

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Molly S. cc
Molly S.
Gordon
Gordon
Gordon espanol spanish
Gordon
Problem  4
practice icon

Simplify \(\sqrt{16a^2b^2}\), assume \(a\geq 0\) and \(b\geq 0\).

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Molly S. cc
Molly S.
Gordon cc
Gordon
Cynthia espanol spanish
Cynthia
Problem  5
practice icon

Simplify \(\sqrt[3]{125a^3}\)

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Preston cc
Preston
Molly S. cc
Molly S.
David espanol spanish
David
Problem  6
practice icon

Simplify \(\sqrt{x^6}\), assume \(x\geq 0\).

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Molly S. cc
Molly S.
Preston cc
Preston
Cynthia espanol spanish
Cynthia
Problem  7
practice icon

Use a calculator to find decimal approximations of each of the following, round to the nearest thousandth.

  1. \(\sqrt{12}\)

  2. \(2\sqrt{3}\)

  3. \(\sqrt{45}\)

  4. \(3\sqrt{5}\)

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Molly S. cc
Molly S.
Gordon
Gordon
Gordon espanol spanish
Gordon
Problem  8

Simplify each expression.

  1. \(\sqrt{36}+\sqrt{64}\)

  2. \(\sqrt{36+64}\)

  3. \(\dfrac{3\sqrt{5}}{6}\)

  4. \(\dfrac{4+6\sqrt{3}}{4}\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  9

A tent pole is \(8\) feet long and makes a \(90^{\circ}\) angle with the ground. One end of a rope is attached to the top of the pole, and the other end is anchored to the ground \(6\) feet from the bottom of the pole. Find the length of the rope.

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Lauren cc
Lauren
Aaron cc
Aaron
Problem  10

Additional Problems
Simplify each expression

  1. \(\displaystyle\frac{3\sqrt{5}}{6}\)

  2. \(\displaystyle\frac{4\sqrt{2}}{2}\)

  3. \(\displaystyle\frac{4+6\sqrt{3}}{4}\)

  4. \(\displaystyle\frac{10+5\sqrt{6}}{5}\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Stefanie cc
Stefanie
Gordon espanol spanish
Gordon
Mini Lecture

Mini Lecture
Simplify.

  1. \(\sqrt{9}\)

  2. \(-\sqrt{9}\)

  3. \(\sqrt{-25}\)

  4. \(-\sqrt{64}\)

  5. \(\sqrt[3]{-8}\)

  6. \(\sqrt{9x^2}\)

  7. \(\sqrt{36a^6}\)

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Mr. McKeague cc
Mr. McKeague
Study Skill

Enrichment
Constructing a Golden Rectangle from a square of side length \(2\).

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Mr. McKeague cc
Mr. McKeague

2
Simplified Form and Properties of Radicals

Problem  1
practice icon

Simplify \(\sqrt{20}\)

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
David espanol spanish
David
Problem  2
practice icon

Simplify \(\sqrt{75}\)

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
David espanol spanish
David
Problem  3
practice icon

Simplify \(\sqrt{25x^3}\)

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
CJ cc
CJ
David espanol spanish
David
Problem  4
practice icon

Simplify \(\sqrt{18y^4}\), assume \(y\geq 0\).

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
David espanol spanish
David
Problem  5
practice icon

Simplify \(3\sqrt{32}\)

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
David espanol spanish
David
Problem  6
practice icon

Simplify \(\sqrt[3]{24x^3}\)

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
David espanol spanish
David
Problem  7
practice icon

Simplify \(\sqrt{\displaystyle\frac{49}{81}}\)

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
David espanol spanish
David
Problem  8
practice icon

Simplify \(\sqrt[4]{\displaystyle\frac{81}{16}}\)

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
David espanol spanish
David
Problem  9

Simplify \(\sqrt{\displaystyle\frac{50}{49}}\)

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Brooke cc
Brooke
Matt cc
Matt
Anthony espanol spanish
Anthony
Problem  10

Simplify \(\sqrt{\displaystyle\frac{12x^2}{25}}\), assume \(x\geq 0\).

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
Anthony espanol spanish
Anthony
Problem  11

Simplify each expression

  1. \(\displaystyle\frac{\sqrt{12}}{6}\)

  2. \(\displaystyle\frac{5\sqrt{18}}{15}\)

  3. \(\displaystyle\frac{6+\sqrt{8}}{2}\)

  4. \(\displaystyle\frac{4+\sqrt{36}}{2}\)

  5. \(\displaystyle\frac{-2+\sqrt{48}}{4}\)

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Stephanie
Stephanie
Spencer
Spencer
Problem  12

Simplify each expression

  1. \(\displaystyle\frac{-4-\sqrt{40}}{4}\)

  2. \(\displaystyle\frac{-6+\sqrt{48}}{2}\)

  3. \(\displaystyle\frac{6-\sqrt{108}}{18}\)

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Molly S. cc
Molly S.
Gordon
Gordon
Gordon espanol spanish
Gordon
Problem  13

Construct a golden rectangle from a square of side length \(4\). Then show that the ratio of the length to the width is the golden ratio \(\displaystyle\frac{1+\sqrt{5}}{2}\).

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Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  14

Additional Problems
Simplify each expression

  1. \(\displaystyle\frac{\sqrt{12}}{6}\)

  2. \(\displaystyle\frac{5\sqrt{18}}{15}\)

  3. \(\displaystyle\frac{6+\sqrt{8}}{2}\)

  4. \(\displaystyle\frac{-1+\sqrt{45}}{2}\)

  5. \(\displaystyle\frac{4+\sqrt{36}}{2}\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Mini Lecture

Mini Lecture
Simplify.

  1. \(\sqrt{12}\)

  2. \(\sqrt[3]{24}\)

  3. \(5\sqrt{80}\)

  4. \(3\sqrt{50xy^2}\)

  5. \(\sqrt{\displaystyle\frac{128}{49}}\)

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Mr. McKeague cc
Mr. McKeague
Study Skill

Roots and Radicals 5: Constructing a Golden Rectangle from a Square of Side 4


No matter how large or small a golden rectangle is, the ratio of length to width is always the same.

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Mr. McKeague cc
Mr. McKeague

3
Addition and Subtraction of Radical Expressions

Problem  1

Combine \(3\sqrt{5}-7\sqrt{5}\)

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Arielle cc
Arielle
Ana
Ana
Preston
Preston
Anthony espanol spanish
Anthony
Problem  2

Combine \(7\sqrt{2}-3\sqrt{2}+6\sqrt{2}\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
David espanol spanish
David
Problem  3

Simplify: \(6\sqrt{5}+5\sqrt{2}-3\sqrt{5}\).

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  4

Combine \(3\sqrt{50}+2\sqrt{32}\)

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Mr. McKeague cc
Mr. McKeague
Betsy
Betsy
Preston
Preston
Anthony espanol spanish
Anthony
Problem  5

Combine terms in the expression \(\sqrt[3]{375}+\sqrt[3]{81}-6\sqrt[3]{3}\).

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  6

Simplify \(a\sqrt{12}+5\sqrt{3a^2}\), assume \(a\geq0\).

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Betsy
Betsy
Molly S. cc
Molly S.
Preston
Preston
Edwin espanol spanish
Edwin
Problem  7

Combine \(\sqrt{20x^3}-3x\sqrt{45x}+10\sqrt{25x^2}\), assume \(x\geq0\).

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
David espanol spanish
David
Problem  8

Simplify \(\displaystyle\frac{6+\sqrt{12}}{4}\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Preston cc
Preston
Anthony espanol spanish
Anthony
Problem  9

Mini Lecture
Simplify.

  1. \(3\sqrt{2}+4\sqrt{2}\)

  2. \(\sqrt{8}+2\sqrt{2}\)

  3. \(\displaystyle\frac{1}{5}\sqrt{75}-\displaystyle\frac{1}{2}\sqrt{12}\)

  4. \(6\sqrt{48}-2\sqrt{12}+5\sqrt{27}\)

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Mr. McKeague cc
Mr. McKeague

4
Multiplication of Radical Expressions

Problem  1

Multiply: \(\sqrt{2}\cdot\sqrt{14}\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  2

Multiply \(\left(3\sqrt{5}\right)\left(2\sqrt{7}\right)\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
David espanol spanish
David
Problem  3

Multiply \(\sqrt{5}\left(\sqrt{2}+\sqrt{5}\right)\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
Anthony espanol spanish
Anthony
Problem  4

Use the power property for radicals to simplify the following radical expressions: \(\left(\sqrt{11}\right)^2\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  5

Use the power property for radicals to simplify the following radical expression: \(\left(\sqrt[3]{-6}\right)^3\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  6

Use the power property for radicals to simplify the following radical expression: \(\left(\sqrt[4]{2}\right)^4\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  7

Use the power property for radicals to simplify the following radical expression: \(\left(\sqrt[5]{3}\right)^5\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  8

Multiply: \(3\sqrt{2}\left(2\sqrt{5}+5\sqrt{6}\right)\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  9

Multiply \(\left(\sqrt{5}+2\right)\left(\sqrt{5}+7\right)\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
Anthony espanol spanish
Anthony
Problem  10

Multiply \(\left(\sqrt{x}+3\right)\left(\sqrt{x}-7\right)\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
David espanol spanish
David
Problem  11

Simplify \(\left(\sqrt{3}-2\right)^2\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
Anthony espanol spanish
Anthony
Problem  12

Multiply: \(\left(\sqrt{6}+\sqrt{3}\right)^2\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  13

Multiply \(\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
Edwin espanol spanish
Edwin
Problem  14

Multiply \(\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)\)

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Betsy
Betsy
Molly S. cc
Molly S.
Preston
Preston
David espanol spanish
David
Problem  15

Multiply: \(\sqrt[3]{9}\cdot\sqrt[3]{3}\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  16

Multiply: \(\left(\sqrt[3]{4}-5\right)^2\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor

5
Division of Radical Expressions

Problem  1

Divide \(\dfrac{\sqrt{60}}{\sqrt{3}}\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  2

Divide: \(\dfrac{6\sqrt{20}}{2\sqrt{5}}\)

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Betsy cc
Betsy
Molly S. cc
Molly S.
Preston cc
Preston
David espanol spanish
David
Problem  3

Divide: \(\dfrac{5\sqrt[3]{48}}{4\sqrt[3]{2}}\)

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Stephanie
Stephanie
Spencer
Spencer
Breylor
Breylor
Problem  4

Simplify \(\sqrt{\displaystyle\frac{1}{2}}\)

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Betsy cc
Betsy
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  5

Simplify \(\sqrt{\displaystyle\frac{2}{3}}\)

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Betsy cc
Betsy
Molly S. cc
Molly S.
Preston cc
Preston
Cynthia espanol spanish
Cynthia
Problem  6

Simplify: \(\sqrt{\dfrac{4x^3y^2}{3z}}\)

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Betsy cc
Betsy
Molly S. cc
Molly S.
Preston cc
Preston
Cynthia espanol spanish
Cynthia
Problem  7

Rationalize the denominator \(\dfrac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}\)

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Molly S. cc
Molly S.
Preston
Preston
Edwin espanol spanish
Edwin
Problem  8

Rationalize the denominator \(\dfrac{2}{5-\sqrt{3}}\)

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Molly S. cc
Molly S.
Preston
Preston
Anthony espanol spanish
Anthony
Problem  9

Rationalize the denominator in the expression \(\dfrac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)

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Mr. Hampton
Mr. Hampton
Winston
Winston
Breylor
Breylor
Stephanie
Stephanie
Problem  10

Simplify each expression.

  1. \(\sqrt[3]{\dfrac{2}{3}}\)

  2. \(\sqrt[3]{\dfrac{3x^4}{4y^2z}}\)

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Mr. Hampton
Mr. Hampton
Stephanie
Stephanie
Breylor
Breylor
Problem  11

Mini Lecture
Simplify.

  1. \(\left(2\sqrt{3}\right)\left(5\sqrt{7}\right)\)

  2. \(\sqrt{3}\left(2\sqrt{2}+\sqrt{3}\right)\)

  3. \(\left(\sqrt{x}+3\right)^2\)

  4. \(\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)\)

  5. Divide \(\dfrac{\sqrt{3}}{\sqrt{5}-\sqrt{2}}\)

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Mr. McKeague cc
Mr. McKeague
Problem  12

Enrichment
Every Golden Rectangle contains an infinite number of other Golden Rectangles. The key to seeing this is rationalizing the denominator.

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Mr. McKeague cc
Mr. McKeague
Problem  13

Roots and Radicals 8: The Golden Ratio in Bagdad in 1150.

Here is one problem that has almost everything we know about roots and radicals contained within it.

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Mr. McKeague cc
Mr. McKeague

6
Equations Involving Radicals

Problem  1

Solve \(\sqrt{x+1}=7\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
David espanol spanish
David
Problem  2

Solve \(\sqrt{2x-3}=-9\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
David espanol spanish
David
Problem  3

Solve \(\sqrt{3a-2}+3=5\)

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Preston
Preston
Molly S. cc
Molly S.
Betsy
Betsy
David espanol spanish
David
Problem  4

Solve \(\sqrt{x+15}=x+3\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Preston
Preston
Anthony espanol spanish
Anthony
Problem  5

Solve \(\sqrt{2x+1}=\sqrt{x-4}+3\)

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Mr. Hampton
Mr. Hampton
Stephanie
Stephanie
Breylor
Breylor
Problem  6

Solve: \(\sqrt[3]{3-4x}+2=5\)

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Mr. Hampton
Mr. Hampton
Stephanie
Stephanie
Breylor
Breylor
Problem  7

Enrichment
Understanding the squaring property of equality.

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Mr. McKeague cc
Mr. McKeague
Mini Lecture

Mini Lecture
Solve.

  1. \(\sqrt{x+1}=2\)

  2. \(\sqrt{x-9}=-6\)

  3. \(\sqrt{3x+4}-3=2\)

  4. \(\sqrt{x+3}=x-3\)

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Mr. McKeague cc
Mr. McKeague