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 108
Chapter 1, Problem 108

Simplify by reducing the index of the radical. x4y812\(\sqrt\)[12]{x^4y^8}

Verified step by step guidance
1
Identify the expression inside the radical: \(\sqrt[12]{x^4 y^8}\), which is a 12th root of the product \(x^4 y^8\).
Recall that the 12th root of a variable raised to a power can be rewritten using rational exponents: \(\sqrt[12]{x^4 y^8} = (x^4 y^8)^{\frac{1}{12}}\).
Apply the exponent to each factor inside the parentheses separately: \((x^4)^{\frac{1}{12}} \cdot (y^8)^{\frac{1}{12}}\).
Simplify the exponents by multiplying powers: \(x^{\frac{4}{12}} \cdot y^{\frac{8}{12}}\).
Reduce the fractions in the exponents to their simplest form: \(x^{\frac{1}{3}} \cdot y^{\frac{2}{3}}\), which is the simplified expression with reduced radical index.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Radical Expressions and Indices

A radical expression involves roots, such as square roots or twelfth roots, indicated by an index. The index shows the root degree, for example, the 12th root means raising a number to the power of 1/12. Understanding how to interpret and manipulate these indices is essential for simplifying radicals.
Recommended video:
Guided course
05:45
Radical Expressions with Fractions

Exponent Rules and Fractional Exponents

Exponents can be expressed as fractions to represent roots, where the denominator is the root index. For example, the 12th root of x^4 can be written as x^(4/12). Applying exponent rules, such as reducing fractions and multiplying exponents, helps simplify expressions with radicals.
Recommended video:
Guided course
7:39
Introduction to Exponent Rules

Simplifying Radicals by Reducing the Index

Reducing the index of a radical means rewriting the radical with a smaller root index by factoring exponents and extracting powers. This process often involves dividing exponents by the greatest common divisor with the index, simplifying the radical to an equivalent but simpler form.
Recommended video:
Guided course
5:48
Adding & Subtracting Unlike Radicals by Simplifying
Related Practice
Textbook Question

Simplify each rational expression. Also, list all numbers that must be excluded from the domain. [x^3+2x^2]/[x+2]

260
views
Textbook Question

Insert either <, >, or = in the shaded area to make a true statement. 3040−34 □ 1415⋅1514\(\frac{30}{40}\) - \(\frac{3}{4}\) \ \(\square\) \ \(\frac{14}{15}\) \(\cdot\) \(\frac{15}{14}\)

749
views
Textbook Question

Simplify by reducing the index of the radical. x6y39\(\sqrt\)[9]{x^6y^3}

922
views
Textbook Question

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

145
views
Textbook Question

Factor and simplify each algebraic expression.[12x−12+6x−32][12x^{-\(\frac\)12}+6x^{-\(\frac\)32}]

1063
views
Textbook Question

In Exercises 103–114, factor completely. x4−5x2y2+4y4

739
views