Unit 6 Radical Functions Homework 1 Nth Roots Simplifying Radicals

Unit 6: Radical Functions Homework 1 – nth Roots and Simplifying Radicals

This comprehensive guide delves into the intricacies of Unit 6, focusing on simplifying radicals and understanding nth roots. We’ll cover essential concepts, provide step-by-step examples, and offer helpful strategies to master this crucial area of algebra. This detailed explanation will equip you to tackle any radical simplification problem with confidence.

Understanding nth Roots

Before diving into simplification, let's solidify our understanding of nth roots. The nth root of a number 'x' is a number 'a' such that aⁿ = x. This can be represented as:

ⁿ√x = a

where:

• n is the index (the small number outside the radical symbol). It represents the root being taken (e.g., square root, cube root, fourth root, etc.).
• x is the radicand (the number under the radical symbol).
• a is the nth root of x.

Examples:

• ²√9 = 3 because 3² = 9 (square root)
• ³√8 = 2 because 2³ = 8 (cube root)
• ⁴√16 = 2 because 2⁴ = 16 (fourth root)
• ⁵√32 = 2 because 2⁵ = 32 (fifth root)

Important Note: When the index 'n' is even, and the radicand 'x' is positive, there are two real nth roots (one positive and one negative). However, the radical symbol (√) generally denotes only the principal (positive) root. For example, ²√9 = 3, not ±3. If we want both roots, we use the ± symbol explicitly (e.g., ±√9 = ±3). When 'n' is odd, there's only one real nth root.

Dealing with Negative Radicands

The behavior of negative radicands depends heavily on the index:

• Even index: The nth root of a negative number is not a real number. For example, ²√-9 is not a real number because no real number squared equals -9. This leads to the concept of imaginary numbers, which are beyond the scope of this basic introduction.
• Odd index: The nth root of a negative number is a negative real number. For instance, ³√-8 = -2 because (-2)³ = -8.

Simplifying Radicals

Simplifying radicals involves expressing a radical expression in its simplest form. This usually means removing any perfect nth power factors from the radicand. Here's a step-by-step approach:

1. Prime Factorization: Break down the radicand into its prime factors. This is fundamental to identifying perfect nth powers.

2. Identify Perfect nth Powers: Look for groups of 'n' identical prime factors. Each group represents a perfect nth power that can be removed from the radical.

3. Extract Perfect nth Powers: For each group of 'n' identical factors, take one factor out of the radical.

4. Simplify: Combine the factors outside the radical and leave any remaining factors inside.

Examples:

Example 1: Simplifying a Square Root

Simplify √72

1. Prime Factorization: 72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

2. Identify Perfect Squares: We have one pair of 2's (2²) and one pair of 3's (3²).

3. Extract Perfect Squares: √(2³ x 3²) = √(2² x 2 x 3²) = 2 x 3 √2 = 6√2

Therefore, √72 simplifies to 6√2.

Example 2: Simplifying a Cube Root

Simplify ³√54

1. Prime Factorization: 54 = 2 x 3 x 3 x 3 = 2 x 3³

2. Identify Perfect Cubes: We have one group of three 3's (3³).

3. Extract Perfect Cubes: ³√(2 x 3³) = 3³√2 = 3³√2

Therefore, ³√54 simplifies to 3³√2.

Example 3: Simplifying a Fourth Root

Simplify ⁴√48

1. Prime Factorization: 48 = 2 x 2 x 2 x 2 x 3 = 2⁴ x 3

2. Identify Perfect Fourth Powers: We have one group of four 2's (2⁴).

3. Extract Perfect Fourth Powers: ⁴√(2⁴ x 3) = 2⁴√3

Therefore, ⁴√48 simplifies to 2⁴√3.

Simplifying Radicals with Variables

The process is similar when dealing with variables, but we need to consider the exponents.

Example 4: Simplify √(x⁶y³)

1. Identify Perfect Squares: We can rewrite the expression as √(x⁶y²y)

2. Extract Perfect Squares: √(x⁶y²y) = x³y√y

Therefore, √(x⁶y³) simplifies to x³y√y.

Example 5: Simplify ³√(a⁹b⁵c⁶)

1. Identify Perfect Cubes: We can group the terms as ³√(a⁹b³b²c⁶)

2. Extract Perfect Cubes: ³√(a⁹b³b²c⁶) = a³bc²³√(b²)

Therefore, ³√(a⁹b⁵c⁶) simplifies to a³bc²³√(b²).

Adding and Subtracting Radicals

Radicals can only be added or subtracted if they have the same radicand and the same index. This is similar to combining like terms in algebra.

Example 6: Simplify 3√5 + 2√5 – √5

Since all three terms have √5 as the radicand, we can combine the coefficients: 3 + 2 -1 = 4. Therefore, the simplified expression is 4√5.

Example 7: Simplify 2√7 + 5√3 – √7 + 2√3

We can group like radicals together: (2√7 – √7) + (5√3 + 2√3) = √7 + 7√3. Therefore, the simplified expression is √7 + 7√3.

Multiplying Radicals

To multiply radicals with the same index, multiply the radicands and keep the index the same.

Example 8: √3 x √12 = √(3 x 12) = √36 = 6

Example 9: (2√5)(3√2) = (2 x 3)√(5 x 2) = 6√10

Dividing Radicals

To divide radicals with the same index, divide the radicands and keep the index the same. It's often necessary to rationalize the denominator (remove any radicals from the denominator) to simplify the expression fully.

Example 10: √15 / √5 = √(15/5) = √3

Example 11: (6√18) / (3√2) = (6/3)√(18/2) = 2√9 = 2 x 3 = 6

Rationalizing the Denominator

If a radical expression has a radical in the denominator, we need to rationalize the denominator to express it in a simpler form. This involves multiplying both the numerator and the denominator by a suitable radical expression to eliminate the radical from the denominator.

Example 12: Rationalize 5/√2

Multiply the numerator and denominator by √2: (5/√2) x (√2/√2) = (5√2)/2

Example 13: Rationalize 3/(√5 + 1)

In this case, we multiply by the conjugate of the denominator (√5 - 1):

[3/(√5 + 1)] x [(√5 - 1)/(√5 - 1)] = [3(√5 - 1)]/[(√5 + 1)(√5 - 1)] = [3(√5 - 1)]/(5 - 1) = [3(√5 - 1)]/4 = (3√5 - 3)/4

Solving Equations with Radicals

Equations involving radicals require careful manipulation to isolate the variable. A common method is to raise both sides of the equation to the power of the index to eliminate the radical.

Example 14: Solve √(x + 2) = 3

Square both sides: (√(x + 2))² = 3² => x + 2 = 9 => x = 7

Always check your solutions in the original equation to ensure they are valid (sometimes squaring both sides introduces extraneous solutions).

Example 15: Solve ³√(2x - 1) = 2

Cube both sides: (³√(2x - 1))³ = 2³ => 2x - 1 = 8 => 2x = 9 => x = 4.5

This comprehensive guide provides a thorough foundation in simplifying radicals and understanding nth roots. By mastering these techniques, you'll significantly enhance your algebraic skills and confidently tackle more complex mathematical problems. Remember to practice regularly to solidify your understanding and build proficiency. Continue exploring advanced concepts and applications of radical expressions to further expand your mathematical knowledge.