Problem 22 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{27 y^{6}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: $$\sqrt{27 y^{6}}$$ Answer: The simplified expression is: $$3y^3\sqrt{3}$$

Step 1: Identify the problem

We are asked to simplify the radical expression: $$ \sqrt{27 y^{6}} $$

Step 2: Break down the number inside the square root

Factorize 27 into its prime factors: $$ 27 = 3 \times 3 \times 3 $$

Step 3: Simplify the square root

Any pair of identical factors under a square root simplifies to the factor itself. Here, we have one pair of 3s: $$ \sqrt{3 \times 3 \times 3} = \sqrt{3^3} $$ Since two of the factors of 3 are inside the square root, we can simplify this expression even further: $$ \sqrt{3^3} = 3\sqrt{3} $$

Step 4: Simplify the variable term

Since we have a variable raised to an even power, we can simplify by taking the square root of that power: $$ \sqrt{y^{6}} = y^3 $$

Step 5: Combine the simplified terms

Finally, combine the simplified number and variable term to get the final expression: $$ 3\sqrt{3} \cdot y^3 = 3y^3\sqrt{3} $$

Step 6: Final Answer

The final simplified expression is: $$ 3y^3\sqrt{3} $$

Key Concepts

Prime Factorization

Prime factorization is a method used to write a number as a product of its prime numbers. This is a helpful technique for simplifying radical expressions. Let's consider the number 27, which can be expressed as:

• 27 = 3 × 3 × 3

Notice that 3 is a prime number, which means it can only be divided evenly by 1 and itself. By identifying the prime factors, it becomes easier to simplify square roots or other higher-order roots.

Now, when you see a problem that involves a radical expression with a number inside, the first step to simplify is typically to break it into its prime factors. By doing so, you can easily identify pairs or groups of identical factors to simplify further.

Radical Expressions

Radical expressions often involve a square root, cube root, or some other root expression, representing the inverse operation of exponentiation. In our example, the original radical expression is

• \(\sqrt{27}\)

To simplify radical expressions like this, we use the concept of prime factorization to identify factors that form perfect squares or higher-order roots. For example, the prime factorization of 27 is 3 × 3 × 3.

When simplifying \(\sqrt{27}\), we look for pairs of numbers (since we're dealing with a square root) under the square root to "release" one factor from beneath the square root sign. Hence, a pair of 3s yields one 3 outside the root, resulting in 3\(\sqrt{3}\). Techniques like this help simplify calculations and provide a more manageable form of the expression.

Variables with Exponents

When dealing with variables that have exponents under a radical, we apply the rule that the square root of a term with an even exponent can be simplified by reducing the exponent by half. Let's see how this works for

• \(\sqrt{y^6}\)

The variable \(y^6\) under a square root can be simplified because we have an even exponent. To simplify, divide the exponent by the index of the radical, which in this case is 2 (since it's a square root), resulting in

• \(\sqrt{y^6} = y^{6/2} = y^3\)

This technique allows us to simplify radicals easily when dealing with variables, making calculations more straightforward and the resulting expressions simpler to manage.

Combining this understanding with the simplified constants from the prime factorization, our original problem expression \(\sqrt{27y^6}\) simplifies to \(3y^3\sqrt{3}\). This approach reveals how variables with exponents and prime factorization together make radicals manageable.