Problem 12 - Intermediate Algebra

Question

Complete each rule for exponents. \(x^{m / n}=\quad=\sqrt[n]{x^{m}}\)

Step-by-Step Solution

Answer

\(x^{m/n} = \sqrt[n]{x^m}\) is the complete rule for exponents using radicals.

Step 1: Start with the Given Expression

We are given the expression \(x^{m/n}\) and need to express it using a radical notation.

Step 2: Rewrite Using Radical Notation

The expression \(x^{m/n}\) can be rewritten in terms of radicals. This involves recognizing that the denominator \(n\) in the exponent signifies a root, and the numerator \(m\) signifies a power. Therefore, it translates to taking the \(n\)-th root of \(x\) and raising the result to the power of \(m\).

Step 3: Apply the Radical Rule

Apply the rule for fractional exponents, which states that \(x^{m/n} = (x^m)^{1/n} = \sqrt[n]{x^m}\). This means that the fraction \(m/n\) in the exponent indicates you take the \(n\)-th root and raise it to the \(m\)-th power.

Step 4: Combine and Simplify

Combine the components into one expression: \(x^{m/n} = \sqrt[n]{x^m}\). This shows that the original expression is equivalent to taking the \(n\)-th root of \(x\) raised to the \(m\) power. Make sure all parts of the expression reflect this transformation accurately.

Key Concepts

Rules of Exponents

Exponents are like shorthand for multiplying a number by itself. When we talk about the rules of exponents, we are discussing the mathematical guidelines for performing operations involving exponents easily.
Understanding these rules is crucial when dealing with expressions in algebra, especially as they appear in various forms.

Here are some key rules:

• Product of Powers: When you multiply two terms with the same base, you can add their exponents. For example, \(x^a \times x^b = x^{a+b}\).
• Power of a Power: When you raise an exponent to another exponent, you multiply the exponents. This means \((x^a)^b = x^{a \times b}\).
• Quotient of Powers: If you're dividing two terms with the same base, you subtract the exponents. For instance, \(\frac{x^a}{x^b} = x^{a-b}\).
• Zero Exponent: Any non-zero number raised to the power of zero equals one, or \(x^0 = 1\).
• Negative Exponent: A negative exponent means you take the reciprocal of the base and turn the exponent positive. For example, \(x^{-a} = \frac{1}{x^a}\).

These rules make calculations with exponents more manageable and are essential for further understanding expressions like \(x^{m/n}\).
When working with exponents using these foundational rules, complex problems can be simplified dramatically.

Radical Notation

Radical notation is another way of expressing roots, which are the inverse operations of exponents. Instead of writing a number with a fractional exponent, we can express it using a radical sign \(\sqrt{}\). This notation is particularly useful because it provides a clear visual representation of the root being taken.

Let's dive deeper into the terms for radical notation:

• Radical Sign \(\sqrt{}\): This is the symbol used to denote the root. It acts as a visual cue that a root operation is being performed.
• Radicand: The number under the radical sign. In our example, if we have \(\sqrt[n]{x}\), \(x\) is the radicand and represents the number whose root is being found.
• Index: The small number above and to the left of the radical sign. This number tells us which root is being taken. For example, \(\sqrt[3]{x}\) refers to the cube root of \(x\).

Understanding radical notation simplifies expressions and allows for easy conversion between radical form and fractional exponent form. This is especially helpful when solving equations, as each form has its particular advantages in different contexts.

Fractional Exponents

Fractional exponents might seem complex, but they hold immense power to simplify expressions and functions. Instead of resolving complicated radicals, fractional exponents let us handle such roots algebraically.

Here's how fractional exponents work:

• A fractional exponent \(x^{m/n}\) means two operations—finding the \(n\)-th root of \(x\) and raising the result to the power \(m\). This combines roots and exponents into a single operation.
• Fractional exponents efficiently communicate what radical notation could express, but in a way more suited for algebraic manipulation.
• For instance, \(x^{1/2} = \sqrt{x}\) and \(x^{3/2} = \sqrt{x^3}\).

The beauty of fractional exponents lies in their versatility. They can interchangeably represent roots and powers, merging neatly with algebraic expressions.
By mastering fractional exponents, you become adept at converting between radical and power expressions, enhancing flexibility in tackling various mathematical problems.