Problem 74 - Algebra and Trigonometry

Question

Simplify the radical expressions in Exercises \(67-74\) if possible. $$ \frac{\sqrt[4]{162 x^{5}}}{\sqrt[4]{2 x}} $$

Step-by-Step Solution

Answer

The simplified form of the radical expression \(\frac{\sqrt[4]{162 x^{5}}}{\sqrt[4]{2 x}}\) is \(3x\).

Step 1: Reduce Radicand to Prime Factors

Begin by reducing each radicand to its prime factors. The prime factorization of \(162\) is \(2 \times 3^{4}\) and \(x^{5}\) can remain as is. For the denominator \(2\) is already a prime number and \(x\) remains as it is. Therefore, the expression becomes:\[ \frac{\sqrt[4]{2 \times 3^{4} x^{5}}}{\sqrt[4]{2x}}\]

Step 2: Simplify Radicals

Next, simplify the radicals knowing that the fourth root of a number is equivalent to raising that value to the ^(1/4) power. The expression simplifies to: \[ \frac{3^{4(1/4)} x^{5(1/4)}}{2^{(1/4)} x^{(1/4)}}\]

Step 3: Simplify the Fraction

Finally, simplify the entire fraction. Notice that the \(2^{(1/4)}\) in the numerator and denominator cancel each other, as well as \(x^{(1/4)}\) in the numerator and denominator. This results in the final simplified expression: \[ 3 x\]

Key Concepts

Prime Factorization

Prime factorization is a crucial initial step when simplifying radical expressions. It involves breaking down a composite number into the product of its prime factors, which are numbers that are only divisible by 1 and themselves. For example, the number 162 can be factored into prime numbers as follows:

• Divide 162 by the smallest prime number 2: 162 ÷ 2 = 81.
• 81 is not divisible by 2 but it is divisible by 3, so we divide it by 3: 81 ÷ 3 = 27.
• 27 is also divisible by 3: 27 ÷ 3 = 9.
• Continue with 9 which is 3 × 3.
• Thus, the prime factorization of 162 is 2 × 3 × 3 × 3 × 3 or, in exponential form, 2 × 3^4.

When factors are not immediately apparent, a prime factorization tree or trial division can aid in systematically reducing a number into its prime components. The ability to identify prime factors is essential for the simplification of radicals because it allows us to pair factors according to the index of the radical (in this case, the fourth root) for easier simplification.

Radicals Simplification

Simplifying radical expressions can seem daunting, but understanding the properties of exponents and radicals can make it much easier. To simplify a radical, one should look to reduce the radicand (the number within the radical sign) to its prime factors and then apply the rule that the nth root of a number raised to a power can be simplified if the exponent is a multiple of the index. In the context of our example, the fourth root of 3^4 simplifies because we can take the fourth root of each prime factor that has a power of 4. Additionally, when simplifying radicals that involve variables, the same concept applies.

For instance, in the expression \(\frac{\sqrt[4]{162 x^{5}}}{\sqrt[4]{2 x}}\), we applied prime factorization to 162 and got 2 × 3^4. Then, recognizing that \(\sqrt[4]{3^4}\) is 3 and \(\sqrt[4]{x^4}\) is x, we dramatically reduce the complexity of the original radical expression. Furthermore, if there are variables to a power greater than the index or common factors in the numerator and denominator of a radical fraction, these can be simplified by dividing the exponents by the index or canceling common factors, respectively. The result is a much simpler expression that is easier to interpret and use in further calculations.

Exponents and Radicals

The interplay between exponents and radicals is at the heart of understanding how to simplify radical expressions. When dealing with radicals, it's often helpful to think in terms of exponents. The nth root of a number can be represented as that number raised to the power of 1/n. For instance, the fourth root (\(\sqrt[4]{•}\)) is equivalent to raising a number to the 1/4 power. This relationship helps in simplifying expressions with radicals, as seen in the process of simplifying the example expression \(\frac{\sqrt[4]{162 x^{5}}}{\sqrt[4]{2 x}}\).

The concept is that any factor raised to a power that is a multiple of the index can be taken out of the radical. Equivalently, when simplifying a radical expression that involves a fraction, factors that are identical in both the numerator and denominator can often be simplified, akin to reducing a fraction to its simplest form. It is essential to remember that this rule applies not just to numbers but to algebraic expressions containing variables, as variables can also have exponents that are subject to the same rules for simplification. In summary, by utilizing the relationship between exponents and radicals, along with prime factorization, one can methodically approach and simplify even the most complicated radical expressions.