Problem 60 - Algebra A Combined Function

Question

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[4]{162 x^{7} y^{20}} $$

Step-by-Step Solution

Answer

The simplified form is \(3xy^5 \sqrt[4]{2x^3}\).

Step 1: Factor the Radicand

Start by factoring each component inside the radical. For the number 162, factor it into prime factors: 162 = 2 * 81 = 2 * 3^4. For the variables, write the powers explicitly: \(x^7\) and \(y^{20}\).

Step 2: Separate the Fourth Roots

Rewrite the expression by separating the radicand into a product of fourth powers. This means writing:\[\sqrt[4]{162 x^7 y^{20}} = \sqrt[4]{2 \cdot 3^4 \cdot x^7 \cdot y^{20}}\]Separate this as:\[\sqrt[4]{2} \cdot \sqrt[4]{3^4} \cdot \sqrt[4]{x^7} \cdot \sqrt[4]{y^{20}}\]

Step 3: Simplify Each Component

Simplify each fourth root:- \(\sqrt[4]{3^4} = 3\). Since \(3^4\) is a perfect fourth power, its fourth root is 3.- \(\sqrt[4]{x^7} = x^{7/4} = x^{1 + 3/4} = x \cdot x^{3/4}\). Split the power such that you take out the highest power that is a multiple of 4.- \(\sqrt[4]{y^{20}} = y^{20/4} = y^5\). Since \(20/4\) is fully divisible, it simplifies to \(y^5\).- \(\sqrt[4]{2}\) does not simplify since 2 is not a perfect fourth power.Combine these simplifications to give:\[3 \cdot y^5 \cdot x \sqrt[4]{x^3} \cdot \sqrt[4]{2}\]

Step 4: Combine and Arrange the Final Expression

Arrange the expression based on the simplified components, first the coefficients (numerical and variable) and then the remaining roots:The final simplified form is:\[3xy^5 \sqrt[4]{2x^3}\]

Key Concepts

Roots and Radicals

When dealing with roots and radicals, we are often looking to simplify expressions containing them. The number in the root symbol indicates which root we need to find. For example, a square root has a 2, a cube root a 3, and so forth. In this exercise, we deal with a fourth root symbolized as \( \sqrt[4]{} \).
To simplify expressions with roots, we attempt to factor the numbers or variables into smaller pieces that are easier to handle. If any factor inside the radical ends up being a perfect power of the root, it can be simplified.

• Factor numbers into their prime components to check if some powers are multiples of the root.
• Variables with exponents are already in power form, making it easier to adjust for the desired root.

Understanding how roots work is essential for further simplification and facilitates dealing with the subsequent steps effectively.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. This is a vital step in simplifying radicals because it helps to identify whether parts of the number can be simplified when under a root.
Take, for instance, the number 162. Breaking it down:

• Start with the smallest prime number, 2: \(162 \div 2 = 81 \).
• Next, use 3, since 81 is not divisible by 2: \(81 = 3^4 \).
• Thus, the prime factorization of 162 is \(2 \times 3^4 \).

Using prime factorization helps identify the perfect powers that correspond to the radical being simplified and makes it easier to see which factors can "exit" the root.

Variable Exponents

Variable exponents require special attention when included in radicals. Each variable with an exponent inside a root can be seen as a number raised to a power.
For variable exponents:

• Figure out the highest exponent that is a multiple of the required root. This allows you to "extract" part of the variable out of the radical, simplifying the term.
• As an example, with \(x^7\) under a fourth root, we consider \(7 = 4 + 3\). Thus, \(x^7 = x^4 \cdot x^3\). The fourth root of \(x^4\) is \(x\), leaving \(x^{3/4}\) inside.

Approaching variable exponents this way makes it easier to see a clear path to simplification, breaking it down step-by-step.

Expressions Simplification

Expression simplification is about making a complex expression easier to understand or work with. This often involves reducing radicals, performing arithmetic operations, and reorganizing terms.
The process involves:

• Breaking down the expression into simpler components using roots and prime factors.
• Combining similar terms or eliminated terms that simplify completely.
• Consolidating terms outside the radical with those that are resolved.
• For instance, after simplifying each component of the exercise, combine them back to a neat form: \(3xy^5 \sqrt[4]{2x^3}\).

By simplifying expressions, we produce a cleaner, more readable result, making it more manageable for both computations and interpretations.