Solving Complex Radical Expressions: A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of radical expressions. If you've ever felt a little intimidated by those square root symbols and complex calculations, don't worry – you're not alone! This guide will break down three radical expressions step-by-step, making it super easy to understand. We'll tackle expressions involving multiplication and division of radicals, and by the end, you'll be a pro at simplifying these kinds of problems. So, grab your pencils, and let's get started!

a) (5√6) × (2√15) ÷ (15√5)

Let's kick things off with our first expression: (5√6) × (2√15) ÷ (15√5). At first glance, it might seem a bit complicated, but don't fret! We'll break it down into manageable steps. The key here is to remember the basic rules of radical operations. We can multiply and divide radicals much like we do regular numbers, but we need to keep the numbers inside the square roots separate from the coefficients (the numbers outside the square roots).

Step 1: Multiply the first two terms

First, we'll multiply (5√6) by (2√15). To do this, we multiply the coefficients (5 and 2) and the numbers inside the square roots (6 and 15) separately. So, we have:

(5 × 2) × √(6 × 15) = 10√90

Now we've simplified the first part of the expression to 10√90. But we're not done yet! We need to see if we can simplify the radical further.

Step 2: Simplify the radical √90

To simplify √90, we need to find the prime factors of 90. Prime factorization is the process of breaking down a number into its prime number components. For 90, the prime factors are 2, 3, 3, and 5 (since 2 × 3 × 3 × 5 = 90). We can rewrite √90 as:

√(2 × 3 × 3 × 5)

Now, we look for pairs of the same factor inside the square root. We have a pair of 3s, which means we can take a 3 out of the square root:

√(3² × 2 × 5) = 3√(2 × 5) = 3√10

So, √90 simplifies to 3√10. Now we substitute this back into our expression:

10√90 = 10 × 3√10 = 30√10

Step 3: Divide by the last term

Now we have 30√10, and we need to divide it by (15√5). Again, we divide the coefficients and the numbers inside the square roots separately:

(30√10) ÷ (15√5) = (30 ÷ 15) × (√10 ÷ √5)

This simplifies to:

2 × √(10 ÷ 5) = 2√2

Final Answer

So, the final answer for expression a) is 2√2. See? Not so scary when we break it down step by step!

b) (3√12) × (-√8) ÷ (6√6)

Next up, let's tackle expression b): (3√12) × (-√8) ÷ (6√6). This one has a negative sign and slightly different numbers, but we'll use the same principles we learned in the first example.

Step 1: Multiply the first two terms

First, multiply (3√12) by (-√8). Remember to multiply the coefficients and the numbers inside the square roots separately, and pay attention to the negative sign:

(3 × -1) × √(12 × 8) = -3√96

Now we have -3√96. Let's simplify that radical.

Step 2: Simplify the radical √96

Find the prime factors of 96. They are 2, 2, 2, 2, 2, and 3 (since 2 × 2 × 2 × 2 × 2 × 3 = 96). So we can rewrite √96 as:

√(2 × 2 × 2 × 2 × 2 × 3)

Look for pairs of the same factor. We have two pairs of 2s, so we can take them out of the square root:

√(2² × 2² × 2 × 3) = 2 × 2 × √(2 × 3) = 4√6

Substitute this back into our expression:

-3√96 = -3 × 4√6 = -12√6

Step 3: Divide by the last term

Now we have -12√6, and we need to divide it by (6√6):

(-12√6) ÷ (6√6) = (-12 ÷ 6) × (√6 ÷ √6)

This simplifies to:

-2 × 1 = -2

Final Answer

The final answer for expression b) is -2. Great job! We're two for two now.

c) (-2√6) × (1/4 √8) ÷ (3/4 √12)

Let's move on to the final expression, c): (-2√6) × (1/4 √8) ÷ (3/4 √12). This one involves fractions, but the same rules apply. Don't let those fractions scare you!

Step 1: Multiply the first two terms

First, multiply (-2√6) by (1/4 √8). Again, multiply the coefficients and the numbers inside the square roots separately:

(-2 × 1/4) × √(6 × 8) = -1/2 √48

So we have -1/2 √48. Time to simplify that radical.

Step 2: Simplify the radical √48

Find the prime factors of 48. They are 2, 2, 2, 2, and 3 (since 2 × 2 × 2 × 2 × 3 = 48). Rewrite √48 as:

√(2 × 2 × 2 × 2 × 3)

Look for pairs of the same factor. We have two pairs of 2s, so we can take them out:

√(2² × 2² × 3) = 2 × 2 × √3 = 4√3

Substitute this back into our expression:

-1/2 √48 = -1/2 × 4√3 = -2√3

Step 3: Divide by the last term

Now we have -2√3, and we need to divide it by (3/4 √12). Dividing by a fraction can be tricky, but remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 3/4 is 4/3, so we rewrite the division as multiplication:

(-2√3) ÷ (3/4 √12) = (-2√3) × (4/3 √12)

Now, let's simplify √12 before we multiply. The prime factors of 12 are 2, 2, and 3 (since 2 × 2 × 3 = 12), so:

√12 = √(2² × 3) = 2√3

Substitute this back into the expression:

(-2√3) × (4/3 × 2√3) = (-2√3) × (8/3 √3)

Now, multiply the coefficients and the numbers inside the square roots:

(-2 × 8/3) × (√3 × √3) = -16/3 × 3

Step 4: Simplify the result

-16/3 × 3 = -16

Final Answer

The final answer for expression c) is -16. Awesome! You've nailed all three expressions.

Conclusion

Alright, guys! We've successfully solved three complex radical expressions. Remember, the key is to break down the problem into smaller, manageable steps. Always simplify radicals by finding their prime factors, and remember that dividing by a fraction is the same as multiplying by its reciprocal. With a little practice, you'll be simplifying radical expressions like a math whiz! Keep up the great work, and don't hesitate to tackle more challenging problems. You've got this! 🚀