Problem 105 - College Algebra Essentials

Question

In Exercises \(101-108\), simplify by reducing the index of the radical. $$\sqrt[6]{x^{4}}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[6]{x^{4}}\) is \(x^{\frac{2}{3}}\).

Step 1: Identify the Radical and its Components

The given expression is \(\sqrt[6]{x^{4}}\). Here, the index of the radical (root) is 6, and the radicand (the number under the root) is \(x^{4}\).

Step 2: Apply Rules of Indices

Remember that the nthRoot(x^m) can be rewritten as \(x^{\frac{m}{n}}\). So we can rewrite given expression as \(x^{\frac{4}{6}}\).

Step 3: Simplify the Fraction

Next task is to simplify the fraction. Both 4 and 6 have common factor of 2 that we can divide out. So, \(x^{\frac{4}{6}}\) simplifies to \(x^{\frac{2}{3}}\).

Key Concepts

Properties of Exponents

Exponents are an integral part of algebra, serving as a foundation for more advanced concepts such as exponential growth and decay, and solving equations involving radicals. The properties of exponents help to simplify expressions and solve equations efficiently.

• Multiplication Rule: When multiplying like bases, you add the exponents. For example, if you have \(a^m \times a^n\), the result is \(a^{m+n}\).
• Division Rule: Here, you subtract the exponents when dividing like bases: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Rule: When raising an exponent to another power, you multiply the exponents: \((a^m)^n = a^{mn}\).
• Power of a Product Rule: Distribute the power to each factor in the product: \((ab)^m = a^m \cdot b^m\).

Understanding and applying these properties is key when simplifying expressions with radicals, as they are often involved when reworking complicated expressions into simpler forms.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, or higher roots. They're a way of representing repeated multiplication by fractions or roots. Here's how you can simplify these kinds of expressions:

• Identify the Index: This tells you which root is being taken. For example, in \(\sqrt[6]{x^4}\), the index is 6.
• Convert to Exponential Form: This can make manipulation easier. For example, \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\).
• Rationalize when Needed: Sometimes, leaving an expression in radical form makes further calculations simpler, especially when dealing with numerical values.

Being comfortable with radical expressions can give you more flexibility when tackling various algebraic problems, especially those involving fractional exponents.

Fraction Simplification

Simplifying fractions is a crucial skill in mathematics that ensures expressions are in their simplest form. Simplification involves making the numerator and denominator as small as possible while still maintaining the same value.

• Finding the Greatest Common Factor (GCF): This is key to simplifying fractions. It involves identifying the highest number that divides both the numerator and the denominator without leaving a remainder.
• Canceling Common Factors: Once the GCF is found, divide both the numerator and the denominator by this value. For example, \(\frac{4}{6}\) simplifies to \(\frac{2}{3}\) since 2 is the GCF.

By mastering fraction simplification, solving equations, especially those with radicals and exponents, becomes more seamless, leading to more accurate results and clearer understanding.