Problem 47 - Algebra A Combined Function

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt[4]{256 x^{8}} $$

Step-by-Step Solution

Answer

The fourth root of \(\sqrt[4]{256x^8}\) is \(4x^2\).

Step 1: Identify the Root and the Expression

The problem requires finding the fourth root of the expression \(256x^8\). This can be written as \( \sqrt[4]{256x^8} \). The fourth root means we are looking for a number which, when raised to the power of 4, gives \(256x^8\).

Step 2: Simplify the Radicand

First, simplify the radicand \(256x^8\). Notice that \(256\) is a perfect power, specifically \(256 = 4^4\) or \(256 = (2^2)^4 = 2^8\). Therefore, \(256x^8 = (2^8)x^8\).

Step 3: Apply the Power Rule

Apply the power rule for radicals: \(\sqrt[n]{a^m} = a^{m/n}\). Use this to find \(\sqrt[4]{(2^8)x^8}\).This expression becomes \((2^8)^{1/4} \cdot (x^8)^{1/4} \).

Step 4: Simplify Each Component

Simplify each component of the expression: - \((2^8)^{1/4} = 2^{8/4} = 2^2 = 4\). - \((x^8)^{1/4} = x^{8/4} = x^2\).

Step 5: Combine the Results and Write the Final Solution

Now, combine the simplified components to get the fourth root of the original expression. Thus, the solution is \(4x^2\).

Key Concepts

Radicand Simplification

Simplification of the radicand is an important step in simplifying radical expressions. The radicand is the expression contained within the radical symbol, in this case, \( \sqrt[4]{256x^8} \). Before applying any rules for extracting roots, it is useful to simplify the radicand as much as possible.
Understanding that \( 256 \) can be expressed as a power of a smaller number is helpful. Notice that \( 256 = 2^8 \), identifying it as a perfect power. This essentially conveys that \( 256 \) can be broken down into smaller, manageable parts that can fit the root power we are dealing with.

• This involves breaking complex or large numbers into base components.
• Recognizing perfect powers makes simplification easier.
• Detailing this process aids in a clearer application of subsequent rules.

So, by converting \( 256 \) into \( 2^8 \), we can rewrite the radicand as \( (2^8)x^8 \). This will make the application of further rules more straightforward, simplifying your work.

Power Rule for Radicals

Once the radicand is simplified, we can apply the power rule for radicals. This rule is essential for converting radical expressions into exponential form, which makes them easier to handle.
The power rule for radicals states that \( \sqrt[n]{a^m} = a^{m/n} \). This allows us to express roots as fractional exponents.

• Transform radicals into a fractional exponent format.
• Makes complex calculations simpler and more intuitive.
• The rule is widely applicable to both numbers and variable expressions.

Applying this rule to our expression \( \sqrt[4]{(2^8)x^8} \), we get \( (2^8)^{1/4} \cdot (x^8)^{1/4} \). This reformulation sets the stage for easy simplification using exponent manipulation.

Perfect Powers

Understanding perfect powers is crucial as it helps in identifying expressions that can be written as powers of integers. Perfect powers are numbers that can be expressed as exactly the integral power of another integer.
In our example, \( 256 \) is a perfect power as it can be written as \( 2^8 \). Recognizing it as a perfect power, it easily simplifies our calculations when combined with rules of radicals and exponents.

• Allows quick simplification of complex expressions.
• Reduces the workload by avoiding lengthy calculations.
• Enhances efficiency in finding roots.

By viewing \( 256 \) as \( 2^8 \), it empowers you to transform and simplify expressions, drastically cutting down the complexity of the problem.

Exponent Manipulation

The final stage in simplifying complex radical expressions involves manipulating exponents. Exponent manipulation is the process of simplifying expressions by adjusting and calculating exponents.
Exponents can be manipulated by using the power rule to convert radical expressions into forms that are easier to handle. This involves raising the bases to new powers in a straightforward manner.

• Simplifies expressions for easier computation.
• Uses fractional exponents from the power rule to break down higher powers.
• Makes the final combination of terms in the expression smooth and efficient.

In our expression \( (2^8)^{1/4} \cdot (x^8)^{1/4} \), simplifying \( (2^8)^{1/4} \) yields \( 2^2 = 4 \) and \( (x^8)^{1/4} \) gives \( x^2 \). Combining these results, the expression is simplified to \( 4x^2 \), illustrating how exponent manipulation can be used to easily find the root of a complex expression.