Question

Use rational exponents to reduce the index of the radical.\(\sqrt{\sqrt{x^{4}}}\)

Short answer

The simplified form of \(\sqrt{\sqrt{x^{4}}}\) using rational exponents results in \(x\).

Step-by-step solution

Converting Radicals to Rational Exponents

Using the power rule, where the exponent of a radical \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\), convert root expressions to rational exponent expressions. Hence, \(\sqrt{x^{4}}\) becomes \(x^{\frac{4}{2}}\). Additionally, for the square root of the result, \(\sqrt{x^{\frac{4}{2}}}\) becomes \(x^{\frac{\frac{4}{2}}{2}}\).

Simplifying the Expression

Simplify the rational exponent. The rule of simplification is as follows: \(x^{\frac{m}{n}} = x^{m*n}\). Therefore, for \(x^{\frac{\frac{4}{2}}{2}}\), the simplification results in \(x^{1}\), because \(\frac{4}{2}*\frac{1}{2} = 1\).

Final Solution

Having followed the steps above faithfully, the final solution to the problem is \(x^{1}\), which is simply \(x\).

Key concepts

Converting Radicals to Exponents

Understanding how to convert radicals to rational exponents is an essential skill in algebra that simplifies the process of working with complex radical expressions. A radical, which is denoted by the \( \sqrt[n]{x} \) for the nth root of \( x \) can be transformed into an exponent form \( x^{\frac{1}{n}} \). This conversion is based on the principle that a root can be expressed as a power. For example, the square root of \( x^4 \) is written as \( \sqrt{x^4} \) and can be converted to \( x^{\frac{4}{2}} \) because a square root is the 2nd root, hence the notation \( \frac{1}{2} \) in the exponent position.

When we encounter a nested radical, such as \( \sqrt{\sqrt{x^4}} \) in our exercise, we apply the same principle recursively. The inner radical converts to \( x^{\frac{4}{2}} \) and the outer radical then takes the form of \( \sqrt{x^{\frac{4}{2}}} \) which becomes \( x^{\frac{\frac{4}{2}}{2}} \). By familiarizing oneself with this conversion, complex radicals can be managed with relative ease, making way for further simplification.

Simplifying Radical Expressions

Once we've converted radicals to exponent form, the next step is to simplify the radical expressions. Simplifying takes any complex or redundant components in the expression and reduces them to their simplest form. In rational exponent terms, \( x^{\left(\frac{m}{n}\right)} \) where \( m \) and \( n \) are integers, we aim to find a single exponent that represents that expression.

For our specific problem, we have \( x^{\frac{\frac{4}{2}}{2}} \) which looks complicated. To simplify, we multiply the exponents together, leading to \( x^{\frac{4}{2} * \frac{1}{2}} \) which simplifies to \( x^1 \) or just \( x \) because any number to the power of 1 is the number itself. This approach illustrates how simplification can drastically reduce the complexity of an expression and make it manageable for further operations or evaluations.

Power Rule for Exponents

The power rule for exponents is a fundamental rule in mathematics that allows us to handle expressions involving powers of powers and achieve simplification. The rule states that when we raise a power to another power, we multiply the exponents. For instance, \( (x^m)^n = x^{m*n} \).

In the context of our example, \( x^{\frac{4}{2}} \) raised to the \( \frac{1}{2} \) power during the conversion of the radical involves applying the power rule. We multiply the exponents \( \frac{4}{2} \) and \( \frac{1}{2} \) which yields the simplified power of \( x^1 \) or just \( x \) as explained in the previous section. By utilizing the power rule effectively, complex expressions become more accessible and more straightforward to work with, greatly aiding in the simplification process.