Skip to main content
Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. P - Fundamental Concepts of AlgebraProblem 71
Chapter 1, Problem 71

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

Verified step by step guidance
1
Recognize that the given expression involves the product of two cube roots: ³√9 ⋅ ³√6. Using the property of radicals, ³√a ⋅ ³√b = ³√(a⋅b), combine the two cube roots into a single cube root: ³√(9⋅6).
Multiply the numbers inside the cube root: 9 ⋅ 6 = 54. This simplifies the expression to ³√54.
Check if the number inside the cube root, 54, can be factored into a perfect cube and another factor. The prime factorization of 54 is 2 ⋅ 3³.
Separate the cube root into two parts using the property ³√(a⋅b) = ³√a ⋅ ³√b. Here, ³√54 = ³√(3³) ⋅ ³√2.
Simplify the cube root of the perfect cube, ³√(3³), which equals 3. The final simplified expression is 3 ⋅ ³√2.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. They can be simplified by applying properties of exponents and roots, which state that the nth root of a product can be expressed as the product of the nth roots of each factor. Understanding how to manipulate these expressions is crucial for simplification.
Recommended video:
Guided course
05:45
Radical Expressions with Fractions

Properties of Exponents

The properties of exponents govern how to handle expressions involving powers and roots. For instance, the property that states a^(m/n) = n√(a^m) allows us to convert between radical and exponential forms. This is essential for simplifying radical expressions, as it helps in rewriting them in a more manageable form.
Recommended video:
Guided course
04:06
Rational Exponents

Multiplication of Radicals

When multiplying radical expressions, such as cube roots, the product can be simplified by combining the radicands under a single radical. For example, ³√a ⋅ ³√b = ³√(a*b). This property is key to simplifying expressions like ³√9⋅³√6, as it allows for the multiplication of the numbers inside the radicals before taking the root.
Recommended video:
Guided course
05:20
Expanding Radicals
Related Practice
Textbook Question

Simplify by reducing the index of the radical : y36\(\sqrt\)[6]{y^3}

1
views
Textbook Question

Write each number in decimal notation without the use of exponents. −8.17×106−8.17×10^6

2
views
Textbook Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. x3+2x2−9x−18

Textbook Question

In Exercises 67–82, find each product. (3x−y)(2x+5y)

1
views
Textbook Question

In Exercises 67–82, find each product. (7x2 y+1)(2x2 y−3)

Textbook Question

In Exercises 69–82, simplify each complex rational expression. (x/3−1)/(x−3)