Problem 509 - Elementary Algebra

Question

In the following exercises, simplify. \(\sqrt[4]{48 y^{6}}\)

Step-by-Step Solution

Answer

2 \cdot \sqrt[4]{3} \cdot y^{3/2}

Step 1: Prime Factorization

First, find the prime factorization of 48. 48 can be factored into prime numbers as follows:48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3 So, 48 = 2^4 × 3

Step 2: Rewrite the Expression

Rewrite the original expression \(\sqrt[4]{48y^6}\) using the prime factorization of 48.\( \sqrt[4]{2^4 \cdot 3 \cdot y^6} \)

Step 3: Simplify the Radicand

Now, apply the fourth root to each factor inside the radical separately.\( \sqrt[4]{2^4} \cdot \sqrt[4]{3} \cdot \sqrt[4]{y^6} \)

Step 4: Apply the Fourth Root

Simplify each term by applying the fourth root:\( \sqrt[4]{2^4} = 2 \)\( \sqrt[4]{3} \) remains unchanged because 3 cannot be simplified further.\( \sqrt[4]{y^6} = y^{6/4} = y^{3/2} = y^{1.5} \)

Step 5: Combine and Simplify the Result

Multiply the simplified terms together:\( 2 \cdot \sqrt[4]{3} \cdot y^{3/2} \)This is the simplified form of the expression.

Key Concepts

Prime Factorization

Prime factorization is a method of expressing a number as the product of its prime numbers. For example, the prime factorization of 48 means breaking it down to the prime numbers that multiply to give us 48.
Here's how it's done:
1. Start with the number 48.
2. Divide 48 by the smallest prime number, which is 2. We get 24.
3. Keep dividing by 2 until we can't anymore: \(48 = 2 \times 2 \times 2 \times 2 \times 3\).
So, the prime factors of 48 are \(2^4 \times 3\).
Knowing prime factorization helps in simplifying expressions that contain radicals and exponents.

Fourth Roots

The fourth root of a number is what you multiply by itself four times to get that number. It is written as \( \sqrt[4]{x} \).
For example, \( \sqrt[4]{16} = 2 \) because \( 2 \times 2 \times 2 \times 2 = 16 \).
When dealing with variables and exponents in the context of fourth roots, using the rule that \(([a^{m/n}] = (a^m)^{1/n})\), helps simplify:
For instance, \((y^6)^{1/4}\) becomes \((y^{6/4})\) which simplifies to \((y^{3/2})\) or \((y^{1.5})\).

Radical Expressions

Radical expressions are mathematical expressions that involve roots. The most common radical expressions include square roots, cube roots, and fourth roots.
To simplify a radical expression, first use prime factorization to break down the number inside the radical.
Next, apply the radical to both the numerical and variable parts separately. For example, \( \sqrt[4]{48 y^6}\) can be rewritten using prime factorization and exponent rules:
First, we rewrite 48 as \( \sqrt[4]{2^4 \times 3}\).
When you simplify, \( \sqrt[4]{2^4} = 2 \) because \( 2^{4/4} = 2^1 \). The expression then becomes \( 2 \times \sqrt[4]{3} \times y^{3/2}\).

Exponent Rules

Understanding exponent rules is essential for simplifying algebraic expressions, especially when working with radicals.
Some key exponent rules to remember are:

• Product Rule: \( x^a \times x^b = x^{a+b} \)
• Quotient Rule: \( x^a / x^b = x^{a-b} \)
• Power of a Power Rule: \( (x^a)^b = x^{a \times b} \)
• Fractional Exponents: \( x^{a/b} = \sqrt[b]{x^a} \)

Applying these rules can simplify expressions such as \( \sqrt[4]{y^6} \). Here, \( y^6 \) can be written as \( y^{6/4} \) which simplifies to \( y^{1.5}\). These rules make complex expressions easier to manage.