1. Fundamental Concepts of Algebra

Evaluate and Perform Operations with Higher Roots

Radicals and Rational Exponents

Simplifying Roots with Higher indexes

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

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

Rationalize the Denominator with Radicals 

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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