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Intermediate Algebra
Rational Exponents and Radicals

1
Rational Exponents

Problem  1
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What are the two square roots of \(64\)?

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Mr. McKeague cc
Mr. McKeague
Stefanie cc
Stefanie
Spencer cc
Spencer
Julieta cc espanol spanish
Julieta
Problem  2
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Simplify if possible: \(\sqrt[3]{-8}\)

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Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  3
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Simplify if possible: \(\sqrt{-4}\)

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Stefanie cc
Stefanie
Logan cc
Logan
Cynthia espanol spanish
Cynthia
Problem  4
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Simplify if possible: \(-\sqrt{25}\)

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Nathan cc
Nathan
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  5
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Simplify if possible: \(\sqrt[5]{-32}\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  6
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Simplify if possible: \(\sqrt[4]{-81}\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  7
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Simplify as much as possible: \(\sqrt{25a^4b^6}\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  8
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Simplify as much as possible: \(\sqrt[3]{x^6y^{12}}\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  9
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Simplify as much as possible: \(\sqrt[4]{81r^8s^{20}}\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  10
practice icon

Write as a root, then simplify: \(8^{\frac{1}{3}}\)

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Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  11
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Write as root, then simplify: \(36^{\frac{1}{2}}\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  12
practice icon

Write as a root, then simplify: \(-25^{\frac{1}{2}}\)

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Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  13
practice icon

Write as a root, then simplify: \((-25)^{\frac{1}{2}}\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  14
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Write as a root, then simplify: \(\left(\dfrac{4}{9}\right)^{\frac{1}{2}}\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  15
practice icon

Write as a rational exponent and simplify: \(\sqrt[3]{x^6y^{12}}\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  16
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Write as a rational exponent and simplify: \(\sqrt[4]{81r^8s^{20}}\)

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Betsy cc
Betsy
CJ cc
CJ
Anthony espanol spanish
Anthony
Problem  17
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Simplify as much as possible: \(8^{\frac{2}{3}}\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  18
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Simplify as much as possible: \(25^{\frac{3}{2}}\)

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Betsy cc
Betsy
Anthony cc
Anthony
Anthony espanol spanish
Anthony
Problem  19
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Simplify as much as possible: \(9^{-\frac{3}{2}}\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  20
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Simplify as much as possible: \(\left(\dfrac{27}{8}\right)^{-\frac{4}{3}}\)

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Betsy cc
Betsy
Anthony cc
Anthony
Anthony espanol spanish
Anthony
Problem  21
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Simplify as much as possible: \(x^{\frac{1}{3}}\cdot x^{\frac{5}{6}}\)

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CJ cc
CJ
Stefanie cc
Stefanie
Cynthia espanol spanish
Cynthia
Problem  22
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Simplify as much as possible: \(\left(y^{\frac{2}{3}}\right)^{\frac{3}{4}}\)

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Betsy cc
Betsy
Anthony cc
Anthony
Anthony espanol spanish
Anthony
Problem  23
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Simplify as much as possible: \(\dfrac{z^{\frac{1}{3}}}{z^{\frac{1}{4}}}\)

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CJ cc
CJ
Stefanie cc
Stefanie
Cynthia espanol spanish
Cynthia
Problem  24
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Simplify as much as possible: \[\left(\frac{a^{-\frac{1}{3}}}{b^{\frac{1}{2}}}\right)^6\]

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Molly S. cc
Molly S.
Octabio
Octabio
Cynthia espanol spanish
Cynthia
Problem  25
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Simplify as much as possible: \(\dfrac{\left(x^{-3}y^{\frac{1}{2}}\right)^4}{x^{10}y^{\frac{3}{2}}}\)

Choose instructor to watch:
Betsy cc
Betsy
Anthony cc
Anthony
Anthony espanol spanish
Anthony

2
Simplifying Radicals

Problem  1
practice icon

Write \(\sqrt{50}\) in simplified form.

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Stefanie cc
Stefanie
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  2
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Write in simplified form: \(\sqrt{48x^4y^3}\), where \(x\), \(y\geq0\).

Choose instructor to watch:
Betsy cc
Betsy
CJ cc
CJ
Edwin espanol spanish
Edwin
Problem  3
practice icon

Write \(\sqrt[3]{40a^5b^4}\) in simplified form.

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  4
practice icon

Write \(\sqrt{12x^7y^6}\) in simplified form.

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Betsy cc
Betsy
CJ
CJ
Julieta cc espanol spanish
Julieta
Problem  5
practice icon

Write \(\sqrt[3]{54a^6b^2c^4}\) in simplified form.

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Stefanie
Stefanie
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  6
practice icon

Simplify \(\sqrt{\displaystyle\frac{3}{4}}\)

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Betsy cc
Betsy
CJ cc
CJ
Edwin espanol spanish
Edwin
Problem  7
practice icon

Simplify \(\sqrt{\displaystyle\frac{5}{6}}\)

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Stefanie cc
Stefanie
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  8
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Rationalize the denominator in \(\displaystyle\frac{4}{\sqrt{3}}\)

Choose instructor to watch:
Betsy cc
Betsy
CJ cc
CJ
Edwin espanol spanish
Edwin
Problem  9

Rationalize the denominator in \(\displaystyle\frac{2\sqrt{3x}}{\sqrt{5y}}\)

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Julieta cc espanol spanish
Julieta
Problem  10
practice icon

Rationalize the denominator in \(\displaystyle\frac{7}{\sqrt[3]{4}}\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague
Julieta
Julieta
CJ cc
CJ
Julieta cc espanol spanish
Julieta
Problem  11
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Simplify \(\sqrt{\displaystyle\frac{12x^5y^3}{5z}}\)

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Stefanie cc
Stefanie
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  12
practice icon

Simplify, but do not assume the variable represents a positive number. \[\sqrt{9x^2}\]

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Mr. Hampton cc
Mr. Hampton
Betsy cc
Betsy
Stephanie cc
Stephanie
Edwin espanol spanish
Edwin
Problem  13
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Simplify, but do not assume the variable represents a positive number. \[\sqrt{x^3}\]

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  14
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Simplify, but do not assume the variable represents a positive number. \[\sqrt{x^2-6x+9}\]

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Betsy cc
Betsy
CJ cc
CJ
Edwin espanol spanish
Edwin
Problem  15
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Simplify, but do not assume the variable represents a positive number. \[\sqrt{x^3-5x^2}\]

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  16

Simplify: \(\sqrt[3]{(-2)^3}\)

Choose instructor to watch:
Betsy cc
Betsy
CJ cc
CJ
Edwin espanol spanish
Edwin
Problem  17

Simplify: \(\sqrt[3]{\left(-5\right)^3}\)

Choose instructor to watch:
Julieta cc
Julieta
Aaron cc
Aaron
Gordon espanol spanish
Gordon

3
Addition and Subtraction of Radical Expressions

Problem  1
practice icon

Combine \(5\sqrt{3}-4\sqrt{3}+6\sqrt{3}\)

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Betsy cc
Betsy
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  2
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Combine \(3\sqrt{8}+5\sqrt{18}\)

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Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  3
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Combine \(7\sqrt{75xy^3}-4y\sqrt{12xy}\), where \(x\), \(y\geq 0\).

Choose instructor to watch:
Betsy cc
Betsy
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  4
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Combine \(10\sqrt[3]{8a^4b^2}+11a\sqrt[3]{27ab^2}\)

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Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  5
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Combine \(\dfrac{\sqrt{3}}{2}+\dfrac{1}{\sqrt{3}}\)

Choose instructor to watch:
Betsy cc
Betsy
CJ
CJ
Edwin espanol spanish
Edwin
Problem  6

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 \(\displaystyle\frac{1+\sqrt{5}}{2}\).

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin

4
Multiplication and Division of Radical Expressions

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

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Stephanie cc
Stephanie
Sirena cc
Sirena
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  2
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Multiply \(\sqrt{3}\left(2\sqrt{6}-5\sqrt{12}\right)\)

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  3
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Multiply \(\left(\sqrt{3}+\sqrt{5}\right)\left(4\sqrt{3}-\sqrt{5}\right)\)

Choose instructor to watch:
Stephanie cc
Stephanie
Sirena cc
Sirena
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  4
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Expand and simplify: \(\left(\sqrt{x}+3\right)^2\)

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  5
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Expand: \(\left(3\sqrt{x}-2\sqrt{y}\right)^2\)

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Mrs. Mooney cc
Mrs. Mooney
Mrs. Diaz cc
Mrs. Diaz
CJ cc
CJ
Julieta cc espanol spanish
Julieta
Problem  6
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Expand and simplify: \(\left(\sqrt{x+2}-1\right)^2\)

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  7
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Multiply \(\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)\)

Choose instructor to watch:
Sirena cc
Sirena
Stephanie cc
Stephanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  8
practice icon

Divide \(\displaystyle\frac{6}{\sqrt{5}-\sqrt{3}}\) (Rationalize the denominator)

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Julieta cc
Julieta
Stefanie cc
Stefanie
Edwin espanol spanish
Edwin
Problem  9
practice icon

Rationalize the denominator \(\displaystyle\frac{\sqrt{5}-2}{\sqrt{5}+2}\)

Choose instructor to watch:
Mrs. Mooney cc
Mrs. Mooney
Mrs. Diaz cc
Mrs. Diaz
CJ cc
CJ
Julieta cc espanol spanish
Julieta

5
Radical Equations and Functions

Problem  1
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Solve for \(x\): \(\sqrt{3x+4}=5\)

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Molly S. cc
Molly S.
Betsy cc
Betsy
CJ cc
CJ
Octabio espanol spanish
Octabio
Problem  2
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Solve: \(\sqrt{4x-7}=-3\)

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  3
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Solve \(\sqrt{5x-1}+3=7\)

Choose instructor to watch:
Molly S. cc
Molly S.
Betsy cc
Betsy
CJ cc
CJ
Octabio espanol spanish
Octabio
Problem  4
practice icon

Solve: \(t+5=\sqrt{t+7}\)

Choose instructor to watch:
Molly S. cc
Molly S.
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  5
practice icon

Solve: \(\sqrt{x-3}=\sqrt{x}-3\)

Choose instructor to watch:
Octabio cc
Octabio
Molly S. cc
Molly S.
Cynthia espanol spanish
Cynthia
Octabio espanol spanish
Octabio
Problem  6
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Solve: \(\sqrt{x+1}=1-\sqrt{2x}\)

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  7
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Solve: \(\sqrt{x+1}=\sqrt{x+2}-1\)

Choose instructor to watch:
Betsy cc
Betsy
CJ cc
CJ
Julieta cc espanol spanish
Julieta
Problem  8
practice icon

Solve \(\sqrt{2x-10}=\sqrt{x-4}+\sqrt{x-12}\)

Choose instructor to watch:
Molly S. cc
Molly S.
Octabio
Octabio
Cynthia espanol spanish
Cynthia
Problem  9
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Solve: \(\sqrt[3]{4x+5}=3\)

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  10
practice icon

Graph \(y=\sqrt{x}\) and \(y=\sqrt[3]{x}\)

Choose instructor to watch:
Katherine cc
Katherine
Betsy cc
Betsy
CJ cc
CJ
Julieta cc espanol spanish
Julieta

6
Complex Numbers

Problem  1
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Write in terms of \(i\): \(\sqrt{-25}\)

Choose instructor to watch:
Stefanie cc
Stefanie
Betsy cc
Betsy
Julieta cc espanol spanish
Julieta
Problem  2
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Write in terms of \(i\): \(\sqrt{-49}\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  3
practice icon

Write in terms of \(i\): \(\sqrt{-12}\)

Choose instructor to watch:
Betsy cc
Betsy
Stefanie cc
Stefanie
CJ cc
CJ
Julieta cc espanol spanish
Julieta
Problem  4
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Write in terms of \(i\): \(\sqrt{-17}\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  5
practice icon

Simplify as much as possible: \(i^{30}\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Julieta cc espanol spanish
Julieta
Problem  6
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Simplify as much as possible: \(i^{11}\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  7
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Simplify as much as possible: \(i^{40}\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  8
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Find \(x\) and \(y\) if \(3x+4i=12-8yi\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  9
practice icon

Find \(x\) and \(y\) if \((4x-3)+7i=5+(2y-1)i\)

Choose instructor to watch:
Jesse cc
Jesse
Stefanie cc
Stefanie
CJ cc
CJ
Julieta cc espanol spanish
Julieta
Problem  10
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Simplify: \((3+4i)+(7-6i)\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  11
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Simplify: \((7+3i)-(5+6i)\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  12
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Simplify \((5-2i)-(9-4i)\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  13
practice icon

Multiply: \((3-4i)(2+5i)\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  14
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Multiply: \(2i(4-6i)\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  15
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Expand: \((3+5i)^2\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  16
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Multiply: \((2-3i)(2+3i)\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  17
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Divide: \(\dfrac{2+i}{3-2i}\)

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  18
practice icon

Divide: \(\dfrac{7-4i}{i}\)

Choose instructor to watch:
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin