1. Fundamental Concepts of Algebra

Add and Subtract Square Roots

Radicals and Rational Exponents

Example:  Add and Subtract Square Roots Containing Variables

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. 
(b) -3x^-1/3

Simplify each expression. See Example 1. (35m^4n)(-2/7mn^2)

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (27/64)^-4/3

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. y^8/y^12

Quotient Rule for Radicals

Perform the indicated operations. Assume all variables represent positive real numbers. 8√(2x) - √(8x) + √(72x)

Simplify each expression. See Example 1. (5x^2y)(-3x^3y^4)

Calculate each value mentally. (0.25^3)(400^3)

Simplifying Square Roots

Calculate each value mentally. (0.1^3/2)(90^3/2)

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

Evaluate each expression for p=-4, q=8, and r=-10. -(p+2)²-3r / 2-q

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

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∛2/3

Simplify each radical. Assume all variables represent positive real numbers. √24m⁶n⁵

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

Write 27^2/3 in radical form and evaluate.

Perform the indicated operations. Assume all variables represent positive real numbers. 3∜x⁵y - 2x∜xy

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. 100^3/2

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 4r^-3/6r^-6

Calculate each value mentally. (24^2)(0.5^2)

Determine whether each statement is true or false. If false, correct the right side of the equation. (2/3)^-2 = (3/2)^2

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. -a^-3

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∜2/25

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. 
(d) ( 3x )^-1/3

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

Perform the indicated operations. Assume all variables represent positive real numbers. (√2 + 3) (√2 - 3)

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

Simplify each radical. Assume all variables represent positive real numbers. ⁹√∜7³

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. 
(b) ( -3x )^-1/3

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (64^5/3)/(64^4/3)

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

Simplify each expression. See Example 1. (3y^4)(-6y^3)

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∜m/n⁴

Multiplying Variables with Rational Exponents - Basic Example 1

Evaluate each expression for p=-4, q=8, and r=-10. 5r / 2p-3r

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

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. - ∛5/8

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent posi-tive real numbers. p^5/4

Radicals - Notation and Simplifying - Radical and Exponential Notation

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent posi-tive real numbers. -m √2y⁵

Simplify each expression. See Example 1. (4^2)(4^8)

Perform the indicated operations. Assume all variables represent positive real numbers. 2∛3 + 4∛24 - ∛81

Write ∛64 using exponents and evaluate.

Perform the indicated operations. Assume all variables represent positive real numbers. √6(3 + √7)

Radical Notation and Simplifying Radicals

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

Simplify each radical. Assume all variables represent positive real numbers. √192

Write each root using exponents and evaluate. ∜-256

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (k^1/3)/(k^2/3)(k^-1)

Simplify each expression. See Example 1. (9^3)(9^5)

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∛8/x⁴

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

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. (4x)^-2

Perform the operation and/or simplify each of the following. Assume all variables represent positive real numbers. (2 + √3) (2 - √3)

Concept Check: By what number should the numerator and denominator of 1/(∛3 - ∛5) be multiplied in order to rationalize the denominator? Write this fraction with a rationalized denominator.

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (z^3/4)/(z^5/4)(z^-2)

Evaluate each expression. See Example 7. 16^1/4

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

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. √11 • √44

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

Perform the indicated operations. Assume all variables represent positive real numbers. 2∛16 + ∛54

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

Simplify each radical. Assume all variables represent positive real numbers. ⁶√√5³

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (p^4/q)^2

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. 4x^-2

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

Perform the indicated operations. Assume all variables represent positive real numbers. ∛64xy² + ∛27x⁴y⁵

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. √14 • √3pqr

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

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (p - 4) / (√p + 2)

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∛√4

Perform the indicated operations. Assume all variables represent positive real numbers. (√3 + √8)²

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. √2/3x

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. r^7/r^10

Perform the indicated operations. Assume all variables represent positive real numbers. ∜81x⁶y³ - ∜16x¹⁰y³

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

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. 
(c) 3x^-1/3

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. x^12/x^8

Rationalizing the denominator with a radical

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

Simplify each radical. Assume all variables represent positive real numbers. ⁸√5⁴

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 expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (5x)^-2(5x^3)^-3/(5^-2x^-3)^-3

Evaluate each expression. See Example 7. 169^1/2

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)

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. 
(d) 3x^1/3

Evaluate each expression. See Example 7. (-4)^1/2

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

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (-2x^5)^5

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (5a^-1)^4(a^2)^-3

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

Simplify each radical. Assume all variables represent positive real numbers. ⁹√5³

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

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

Match each expression in Column I with its equivalent expression in Column II. Choices may be used once, more than once, or not at all. See Example 3.  a. 6^0 b. -6^0 c. (-6)^0 d. -(-6)^0  A. 0 B. 1 C. -1 D. 6 E. -6

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

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 1/(2 + √5)

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (6^4)^3

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

Simplify each expression. See Example 1. (n^6)(n^4)(n)

Answer each question. If the lengths of the sides of a cube are tripled, by what factor will the volume change?

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

Perform the indicated operations. Assume all variables represent positive real numbers. (3√2 + √3) (2√3 - √2)

Determine whether each statement is true or false. If false, correct the right side of the equation. (3x^2)^-1 = 3x^-2

Determine whether each statement is true or false. If false, correct the right side of the equation. (m^2/3)(m^1/3) = m^2/9

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∜(g³h⁵/9r⁶)

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. 8^2/3

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (√7 - 1) / (2√7 + 4√2)

Write each root using exponents and evaluate. ∛-125

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 4a^5(a^-1)^3/(a^-2)^-2

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (y^7/3)(y^-4/3)

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

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. 
(c) ( 3x )^1/3

Perform operations with rational exponents

Simplify each radical. Assume all variables represent positive real numbers. ∛(25 (-3)⁴ (5)³ )

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 expression. See Example 1. (a^8)(a^5)(a)

Evaluate each expression for p=-4, q=8, and r=-10. q/2-r/3 / 3p/4+q/8

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

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√∛9

Determine whether each statement is true or false. If false, correct the right side of the equation. (y^2)(y^5) = y^7

Evaluate each expression for p=-4, q=8, and r=-10. 3q/r - 5/p

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 4^8/4^6

Square Roots, Cube Roots, and Other Roots

Write each root using exponents and evaluate. ∜256

Multiply rational and irrational numbers

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

Write each root using exponents and evaluate. ∜81

Perform the operation and/or simplify each of the following. Assume all variables represent positive real numbers. √50

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√x² • ⁵√x³

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent posi-tive real numbers. -12x^1/2

Simplify each radical. Assume all variables represent positive real numbers. ⁶√11³

Simplify each radical. Assume all variables represent positive real numbers. ⁶√x¹⁸y²

Find square roots of perfect squares

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. -(2x^0y^4)^3

Perform the operation and/or simplify each of the following. Assume all variables represent positive real numbers. 3√xy - 8√xy

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (p^1/5p^7/10p^1/2)/(p^3)^-1/5

Write each root using exponents and evaluate. ∛-343

Negative Exponents and Fractional Exponents - Examples

Evaluate each expression. See Example 7. (-64/27)^1/3

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

Calculate each value mentally. (20^2/3)/(5^3/2)

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. (1/3)^-2

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

Match each expression in Column I with its equivalent expression in Column II. See Example 8.  a. (4/9)^3/2 b. (4/9)^-3/2 c. -(9/4)^3/2 d. -(9/4)^-3/2  A. 27/8 B. -27/8 C. 8/27 D. -8/27

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

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (3^1/2)(3^3/2)

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

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent posi-tive real numbers. -3 √5p³

Write equivalent radical and rational exponent expressions

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∛ 7x • ∛ 2y

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)

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (z^1/3z^-2/3z^1/6)/(z^-1/6)^3

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

Write each root using exponents and evaluate. ∛125

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

Simplify each radical. Assume all variables represent positive real numbers. - ∜243

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

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. -81^3/4

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. -4r^-2(r^4)^2

Write each root using exponents and evaluate. - ∛-343

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. (-4 /∛3 ) + (1/∛24) - ( 2/∛81)

Write each root using exponents and evaluate. ⁵√32

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

Perform the indicated operations. Assume all variables represent positive real numbers. 3√72m² - 5√32m² - 3√18m²

Simplify each radical. Assume all variables represent positive real numbers. -9 ⁵√243

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

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 16m^-5n^4/12m^2n^-3

Write each root using exponents and evaluate. ∜-81

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

Perform the indicated operations. Assume all variables represent positive real numbers. 4√3(√7 - 2√11)