Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Radical Expressions

College Algebra

0. Review of Algebra

Radical Expressions

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2:14 minutes

Problem 70

Textbook Question

In Exercises 65–74, simplify each radical expression and then rationalize the denominator.5m⁴n⁶√ -------------15m³n⁴

Verified step by step guidance

Step 1: Simplify the expression under the square root. The expression is \( \frac{5m^4n^6}{15m^3n^4} \). Start by simplifying the fraction by dividing the coefficients and subtracting the exponents of like bases.

Step 2: Divide the coefficients: \( \frac{5}{15} = \frac{1}{3} \).

Step 3: Simplify the exponents of \( m \): \( m^{4-3} = m^1 \) or simply \( m \).

Step 4: Simplify the exponents of \( n \): \( n^{6-4} = n^2 \).

Step 5: The simplified expression under the square root is \( \frac{mn^2}{3} \). Now, rationalize the denominator by multiplying the numerator and the denominator by \( \sqrt{3} \) to eliminate the square root in the denominator.

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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, and are often represented with the radical symbol (√). Simplifying radical expressions requires identifying perfect squares or cubes within the radicand (the expression under the root) to reduce the expression to its simplest form. Understanding how to manipulate these expressions is crucial for solving problems involving radicals.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any radicals from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable radical that will result in a rational number in the denominator. This technique is important in algebra to simplify expressions and make them easier to work with.

Exponents and Their Properties

Exponents represent repeated multiplication of a base number and have specific rules that govern their manipulation, such as the product of powers, quotient of powers, and power of a power. In the context of simplifying radical expressions, understanding how to convert between radical and exponent forms is essential, as radicals can be expressed as fractional exponents, facilitating simplification and rationalization.