1. Fundamental Concepts of Algebra

Rationalize Denominators

Radicals and Rational Exponents

Rationalize the Denominator with Radicals 

Quotient Rule for Radicals

Simplifying Square Roots

Simplify by reducing the index of the radical : [y^3]^(1/6)

In Exercises 33–44, add or subtract terms whenever possible. 3√8−√32+3√72−√75

Simplifying radicals - High powered roots

Using Quotient Rule to Write as a Single Radical 

In Exercises 91–100, simplify using properties of exponents. (3y^1/4)^3/y^1/12

In Exercises 33–44, add or subtract terms whenever possible. 4√13x−6√13x

All about DIVIDING SQUARE ROOTS

Simplify each exponential expression in Exercises 23–64. (−4/x)^3

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √48x^3/√3x

Evaluate each exponential expression in Exercises 1–22. 2^2⋅2^3

Simplify Square Roots of Perfect Squares:  Square Root of 81 & (121/144), Negative Square Root of 64

Evaluate each exponential expression in Exercises 1–22. 2^3/2^7

In Exercises 75–82, add or subtract terms whenever possible. √3+³√15

Simplifying Roots with Higher indexes

Multiplying Variables with Rational Exponents - Basic Example 1

In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (3x−4 y z−7)(3x)−3

Radicals - Notation and Simplifying - Radical and Exponential Notation

Radical Notation and Simplifying Radicals

Evaluate each exponential expression in Exercises 1–22. 3^8/3^4

Evaluate each exponential expression in Exercises 1–22. 4^−3

In Exercises 47 - 49, add or subtract terms whenever possible. 7√5 + 13√5

Simplify each exponential expression in Exercises 23–64. x^−5⋅x^10

In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. 30/40−3/4 □ 14/15⋅15/14

Simplify each exponential expression in Exercises 23–64. (3x^4/y)^−3

Evaluate each exponential expression in Exercises 1–22. −2^6

Rationalize the denominator

Simplifying Radical Expressions Involving Variables - Example 2

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √24x^4/√3x

In Exercises 101–108, simplify by reducing the index of the radical. ¹²√x^4y^8

Rationalizing the denominator with a radical

Simplify each exponential expression in Exercises 23–64. x^14/x^7

In Exercises 91–100, simplify using properties of exponents. (25x^4y^6)^1/2

In Exercises 47 - 49, add or subtract terms whenever possible. 4√72 - 2√48

Evaluating Numbers Raised to Fractional Exponents

In Exercises 83–90, evaluate each expression without using a calculator. 16^(−6/2)

Simplify each exponential expression in Exercises 23–64. x^0 y^5

In Exercises 50 - 53, rationalize the denominator. √2/√3

Simplifying Radical Expressions Involving Variables - Example 1

Squares and Square Roots (Simplifying Math)

Rewrite each expression without the absolute value bars. |√2-1|

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √10x⋅√ 8x

In Exercises 50 - 53, rationalize the denominator. 30/√5

Simplify each exponential expression in Exercises 23–64. x^3⋅x^7

Simplify the radical expressions in Exercises 67–74, if possible. ⁵√64x^6/⁵√2x

Simplify the radical expressions in Exercises 67–74, if possible. ³√9⋅³√6

In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. |−20| □ |−50|

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √1/49

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √144+25

In Exercises 91–100, simplify using properties of exponents. (x^2/3)^3

In Exercises 101–108, simplify by reducing the index of the radical. ⁹√x^6 y^3

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. −√36

Perform operations with rational exponents

In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. 8/13÷8/13 □ |−1|

Simplify each exponential expression in Exercises 23–64. x^30/x^−10

Square Roots, Cube Roots, and Other Roots

Multiply rational and irrational numbers

Simplify the radical expressions in Exercises 58 - 62. ∛81

Find square roots of perfect squares

Negative Exponents and Fractional Exponents - Examples

Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ³√8

In Exercises 33–44, add or subtract terms whenever possible. 8√5+11√5

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (x^−5/4y^1/3x^−3/4)^−6

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √500x^3/√10x^−1

In Exercises 45–54, rationalize the denominator. 2/√10

Example:  Add and Subtract Square Roots Containing Variables

Write equivalent radical and rational exponent expressions

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √36

Identify perfect cubes and find cube roots

Simplify each exponential expression in Exercises 23–64. (3x^4)(2x^7)

In Exercises 45–54, rationalize the denominator. 13/(3+√11)

Simplify each exponential expression in Exercises 23–64. 10x^4 y^9/30x^12 y^−3

Evaluating Numbers with Rational Exponents by using Radical Notation - Basic Example 1

In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (2^−1x^−3y^−1)^−2(2x^−6y^4)^−2(9x^3y^−3)^0/(2x^−4y^−6)^2

Simplify each exponential expression in Exercises 23–64. (x^11)^5

Simplify each exponential expression in Exercises 23–64. (3a^−5 b^2/12a^3 b^−4)^0

Simplifying Radical Expressions Involving Variables - Example 3

Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ⁴√−16

Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ⁵√(−3)^5

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √(−17)^2

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √−25

Adding and Subtracting Radicals 

In Exercises 45–54, rationalize the denominator. 11/(√7−√3)

Simplify Square Roots (Perfect Squares)

In Exercises 75–82, add or subtract terms whenever possible. ³√54xy^3−y³√128x

Quotient Property of Square Roots

Simplify each exponential expression in Exercises 23–64. 25a^13 b^4/−5a^2 b^3

Simplify the radical expressions in Exercises 58 - 62. 4∛16 + 5∛2

Simplify expressions involving square roots

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √25−√16

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √6x⋅√3x^2

In Exercises 50 - 53, rationalize the denominator. 14/(√7 - √5)

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (49x^−2y^4)^−1/2(xy^1/2)

What is a Square Root and a Perfect Square?

In Exercises 45–54, rationalize the denominator. 7/(√5−2)

Simplify the radical expressions in Exercises 67–74, if possible. ³√150

Find the square root of a perfect square

Simplify each exponential expression in Exercises 23–64. (4x^3)^−2

Explore volume using radical and rational exponent expressions

Add and Subtract Radicals

Evaluate each exponential expression in Exercises 1–22. (−3)^0 

Simplify each exponential expression in Exercises 23–64. x^−2y

Evaluate each exponential expression in Exercises 1–22. (−2)^6

Adding & Subtracting Radical Expressions

In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (x^−2 y)^−3/(x^2 y^−1)^3

In Exercises 75–82, add or subtract terms whenever possible. 4⁵√2+3⁵√2

In Exercises 101–108, simplify by reducing the index of the radical. ⁴√7^2

Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ⁶√1/64

Evaluate each exponential expression in Exercises 1–22. (3^3)^2

In Exercises 33–44, add or subtract terms whenever possible. √50x−√8x

Multiplying Variables with Rational Exponents - Basic Examples 2

Simplify the radical expressions in Exercises 58 - 62. ∜(32x^5)/∜(16x) (Assume that x > 0.)

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