Simplify Radicals & Calculate Expressions: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of radicals and radical expressions. We'll be simplifying some complex radical expressions and tackling calculations involving various roots. So, grab your notebooks, and let's get started!

1) Simplifying Radicals

Simplifying radicals involves breaking down the radicand (the number inside the root) into its prime factors and then extracting any perfect powers. This makes the radical expression cleaner and easier to work with. Let's tackle the expressions you provided:

c) Simplifying ³√378 - ³√896 + ³√1750

This expression involves cube roots, so we need to find perfect cubes within the radicands. Let's break down each term:

First, let's consider ³√378. We need to find the prime factorization of 378.

378 = 2 × 189 = 2 × 3 × 63 = 2 × 3 × 3 × 21 = 2 × 3 × 3 × 3 × 7 = 2 × 3³ × 7

So, ³√378 = ³√(3³ × 2 × 7) = 3³√(2 × 7) = 3³√14.

Next, let's simplify ³√896. Factoring 896, we have:

896 = 2 × 448 = 2 × 2 × 224 = 2 × 2 × 2 × 112 = 2 × 2 × 2 × 2 × 56 = 2 × 2 × 2 × 2 × 2 × 28 = 2 × 2 × 2 × 2 × 2 × 2 × 14 = 2⁷ × 7 = 2⁶ × 2 × 7 = (2²)³ × 2 × 7

Thus, ³√896 = ³√(2⁶ × 2 × 7) = ³√((2²)³ × 2 × 7) = 2² ³√(2 × 7) = 4³√14.

Now, let's break down ³√1750. Factoring 1750 gives us:

1750 = 2 × 875 = 2 × 5 × 175 = 2 × 5 × 5 × 35 = 2 × 5 × 5 × 5 × 7 = 2 × 5³ × 7

Therefore, ³√1750 = ³√(5³ × 2 × 7) = 5³√(2 × 7) = 5³√14.

Now we can substitute these simplified radicals back into the original expression:

³√378 - ³√896 + ³√1750 = 3³√14 - 4³√14 + 5³√14

Combine the terms:

(3 - 4 + 5)³√14 = 4³√14

So, the simplified expression is 4³√14.

d) Simplifying √1188 - ³√891 + 2⁴√2673

This expression involves square roots, cube roots, and fourth roots, making it a bit more challenging. We'll simplify each term individually.

First, consider √1188. Let's find the prime factorization of 1188:

1188 = 2 × 594 = 2 × 2 × 297 = 2² × 3 × 99 = 2² × 3 × 3 × 33 = 2² × 3 × 3 × 3 × 11 = 2² × 3³ × 11

So, √1188 = √(2² × 3³ × 11) = √(2² × 3² × 3 × 11) = 2 × 3√(3 × 11) = 6√33.

Next, let's simplify ³√891. Factoring 891, we get:

891 = 3 × 297 = 3 × 3 × 99 = 3 × 3 × 3 × 33 = 3 × 3 × 3 × 3 × 11 = 3⁴ × 11

Thus, ³√891 = ³√(3⁴ × 11) = ³√(3³ × 3 × 11) = 3³√(3 × 11) = 3³√33.

Now, let's break down ⁴√2673. Factoring 2673 gives us:

2673 = 3 × 891 = 3 × 3 × 297 = 3 × 3 × 3 × 99 = 3 × 3 × 3 × 3 × 33 = 3 × 3 × 3 × 3 × 3 × 11 = 3⁵ × 11

Therefore, ⁴√2673 = ⁴√(3⁵ × 11) = ⁴√(3⁴ × 3 × 11) = 3⁴√(3 × 11) = 3⁴√33.

Now we substitute these simplified radicals back into the original expression:

√1188 - ³√891 + 2⁴√2673 = 6√33 - 3³√33 + 2(3⁴√33) = 6√33 - 3³√33 + 6⁴√33

Since the radicals are different (√33, ³√33, and ⁴√33), we cannot combine them further. So, the simplified expression is 6√33 - 3³√33 + 6⁴√33.

2) Calculating Radical Expressions

Now, let's move on to calculating expressions involving radicals. We'll use the properties of radicals to simplify and then compute the results.

a) Calculating √18 • √15

To calculate this, we use the property √a • √b = √(a • b). So:

√18 • √15 = √(18 • 15) = √270

Now, simplify √270 by finding its prime factorization:

270 = 2 × 135 = 2 × 3 × 45 = 2 × 3 × 3 × 15 = 2 × 3 × 3 × 3 × 5 = 2 × 3³ × 5

So, √270 = √(3³ × 2 × 5) = √(3² × 3 × 2 × 5) = 3√(3 × 2 × 5) = 3√30.

Therefore, √18 • √15 = 3√30.

b) Calculating √192 : √8

Using the property √a / √b = √(a / b), we have:

√192 : √8 = √(192 / 8) = √24

Now, simplify √24:

24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2³ × 3

Thus, √24 = √(2³ × 3) = √(2² × 2 × 3) = 2√(2 × 3) = 2√6.

So, √192 : √8 = 2√6.

c) Calculating 3√12 • √21

First, let's multiply the radicals using the property √a • √b = √(a • b):

3√12 • √21 = 3√(12 • 21) = 3√252

Now, simplify √252:

252 = 2 × 126 = 2 × 2 × 63 = 2² × 3 × 21 = 2² × 3 × 3 × 7 = 2² × 3² × 7

So, √252 = √(2² × 3² × 7) = 2 × 3√7 = 6√7

Substitute this back into the expression:

3√252 = 3(6√7) = 18√7.

Therefore, 3√12 • √21 = 18√7.

d) Calculating ⁴√16 • ⁴√9 : ⁴√3

Using the properties ⁴√a • ⁴√b = ⁴√(a • b) and ⁴√a / ⁴√b = ⁴√(a / b), we get:

⁴√16 • ⁴√9 : ⁴√3 = ⁴√(16 • 9) / ⁴√3 = ⁴√(16 • 9 / 3) = ⁴√(16 • 3) = ⁴√48

Now, simplify ⁴√48:

48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3

So, ⁴√48 = ⁴√(2⁴ × 3) = 2⁴√3.

Therefore, ⁴√16 • ⁴√9 : ⁴√3 = 2⁴√3.

Conclusion

We've successfully simplified radicals and calculated radical expressions by breaking down radicands into prime factors and using the properties of radicals. Remember, the key is to identify perfect powers within the roots and simplify step by step. Keep practicing, and you'll become a radical master in no time! Keep up the great work, guys!