Problem 42 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[6]{\frac{5 y^{12}}{64}} $$

Step-by-Step Solution

Answer

\( \frac{y^2 \cdot \sqrt[6]{5}}{2} \)

Step 1: Identify the Problem

The given expression is \( \sqrt[6]{\frac{5 y^{12}}{64}} \). Our task is to simplify this sixth root of a fraction that includes a variable and a numerical value.

Step 2: Break Down the Radicand

The radicand (the expression inside the root) is \( \frac{5 y^{12}}{64} \). We can address the numerator and denominator separately for simplification.

Step 3: Simplify the Numerator

The numerator is \( 5 y^{12} \). Since the sixth root applies to each factor within the numerator, we have \( y^{12} \). Taking the sixth root of this, it becomes \( (y^{12})^{1/6} = y^2 \), because \( \frac{12}{6} = 2 \). The number 5 remains unchanged as it cannot be simplified under the sixth root.

Step 4: Simplify the Denominator

The denominator is 64, which can be rewritten as \( 64 = 2^6 \). Taking the sixth root of \( 64 \), we get \( \sqrt[6]{2^6} = 2 \), because the power and root cancel each other.

Step 5: Combine Simplifications

Bringing together the simplifications, we rewrite the original expression as: \[ \frac{\sqrt[6]{5} \cdot y^2}{2} \].

Step 6: Final Expression

The simplified expression is: \[ \frac{y^2 \cdot \sqrt[6]{5}}{2} \]. This is as simplified as it can get since \( \sqrt[6]{5} \) cannot be simplified further without a calculator.

Key Concepts

Understanding the Sixth Root

In mathematics, when we talk about roots, we're generally referring to finding a number which, when raised to a specific power, gives the original number. A sixth root is a special type of root. It finds a number which, when multiplied by itself six times, equals the given number. This is not as common as square roots or cube roots, but it operates on the same principle.

For instance, if we have a number expressed as \( x^6 \), then the sixth root of \( x^6 \) would be \( x \). This is because \( (x^{1/6})^6 = x \).

In our problem, applied to the variable \( y^{12} \), the sixth root can be calculated as \((y^{12})^{1/6} = y^2 \) because the exponent is divided by the root, \( \frac{12}{6} = 2 \). This results in an easy computation and simplification process, removing the need for more complex calculations.

Simplifying Fractions

Fractions are a basic yet crucial part of mathematics, and simplifying them can make calculations a lot easier. Simplifying a fraction means making both the numerator and the denominator as small as possible while keeping the fraction equivalent to the original.

In the context of radical expressions, simplifying fractions often involves factoring the numerator and the denominator. For our example, the fraction in the radicand is \( \frac{5 y^{12}}{64} \). Here, it helps to break down the denominator separately.\( 64 \) is a perfect power in this instance and is expressed as \( 2^6 \). By taking the sixth root, we simplify the denominator from \( 64 \) to \( 2 \) since \( \sqrt[6]{2^6} = 2 \) becomes a simple arithmetic operation.

Numerators may need different steps, as not every number is a perfect power. In our numerator, \( y^{12} \) simplifies through exponents, while the number \( 5 \) remains since it cannot be further simplified by the sixth root unless expressed decimally.

Radicand Simplification

The radicand is the expression under the radical sign and can be a number, a variable, or a combination of both. Simplifying the radicand is critical to simplifying the entire expression.

To simplify a radicand such as \( \frac{5 y^{12}}{64} \), we handle the components separately. Breaking it down helps manage both the numerator and the denominator individually, making the process systematic. Let's simplify such a radicand:

• Numerator: \( 5 y^{12} \) allows us to take the sixth root from the variable part, leaving \( y^2 \), while the constant \( 5 \) remains unaffected.
• Denominator: \( 64 \) simplifies directly to \( 2 \) through \( \sqrt[6]{64} = 2 \), as \( 64 \) is a power of the base number \( 2 \).
  

By understanding how to break down each part of the radicand, we can simplify the entire expression accurately. This often leads to a much neater form as demonstrated in our final simplified result, \( \frac{y^2 \cdot \sqrt[6]{5}}{2} \). Here, the simplification of each component resulted in a more navigable and less cumbersome expression.