Problem 49 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[3]{x y^{2}}\)

Step-by-Step Solution

Answer

\(x^{\frac{1}{3}}y^{\frac{2}{3}}\)

Step 1: Understand the Problem

We are given the expression \(\sqrt[3]{x y^{2}}\) and need to express it using positive rational exponents. A rational exponent \(\frac{m}{n}\) is equivalent to taking the \(n\)-th root and raising to the \(m\)-th power.

Step 2: Apply Rational Exponent Rule

Recognize that \(\sqrt[3]{x y^{2}}\) denotes the cube root of \(xy^2\). The cube root can be represented as raising the expression to the \(\frac{1}{3}\) power: \((x y^{2})^{\frac{1}{3}}\).

Step 3: Distribute the Exponent

Apply the exponent to each component of the expression inside the parentheses. According to exponent laws, \((xy^{2})^{\frac{1}{3}} = x^{\frac{1}{3}}(y^{2})^{\frac{1}{3}}\).

Step 4: Simplify Using Exponent Rules

Now focus on \((y^{2})^{\frac{1}{3}}\). When raising a power to another power, multiply the exponents: \(y^{2\cdot \frac{1}{3}} = y^{\frac{2}{3}}\).

Step 5: Combine the Exponents

The expression with positive rational exponents thus becomes: \(x^{\frac{1}{3}}y^{\frac{2}{3}}\).

Key Concepts

Exponent Laws

Understanding exponent laws is crucial when dealing with expressions that involve powers and roots. These laws provide guidelines that help us manipulate and simplify expressions easily. Here are some important exponent laws to remember:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power Rule: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• Power of a Product Rule: Distribute the exponent to each factor inside the parentheses: \((ab)^n = a^n \cdot b^n\).

These rules are particularly useful when working with rational exponents, allowing us to convert roots into fractional power expressions. Essentially, a root such as the cube root is just another way of expressing the exponent \(\frac{1}{3}\). This knowledge lays a strong foundation for solving various algebraic problems.

Cube Root

The cube root of a number is a special type of root that asks, "What number, when multiplied by itself three times, gives me this original number?" In mathematical notation, it is represented as \( \sqrt[3]{a}\).

When dealing with rational exponents, the cube root can be expressed as the power of one-third: \(a^{\frac{1}{3}}\). This conversion is helpful because it allows us to consistently apply exponent laws to simplify expressions. For example:

\[ \sqrt[3]{x y^{2}} = (x y^{2})^{\frac{1}{3}} \]

After rewriting the root as an exponent, you can easily manipulate the expression using the power of a product rule, breaking it down to

\[ x^{\frac{1}{3}}\cdot(y^{2})^{\frac{1}{3}} \].

This makes cube roots much more manageable when simplifying algebraic expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to interpret and work with in equations.

Here’s a straightforward approach to simplifying expressions with rational exponents:

• Convert roots to exponent form using the rule that \( \sqrt[n]{a} = a^{\frac{1}{n}} \).
• Apply exponent laws, such as distributing exponents across products and using the power of a power rule.
• Combine like terms by adding or subtracting similar bases raised to any power.

In the context of our exercise, we start with \( \sqrt[3]{x y^{2}} \), rewrite it as \( (x y^{2})^{\frac{1}{3}} \), and then simplify by distributing the exponent:

\[ x^{\frac{1}{3}} \cdot y^{\frac{2 \cdot \frac{1}{3}}} = x^{\frac{1}{3}} \cdot y^{\frac{2}{3}} \].

This step-by-step simplification helps yield tidy, workable expressions that are free from roots.

Positive Exponents

Positive exponents are important because they express standard powers without inverses, making calculations straightforward and non-repetitive. Exponents are indicators of how many times a number, known as the base, is multiplied by itself.

In algebra, we often convert expressions to use positive exponents for clarity and ease of computation. For instance, in the exercise, the expression \( \sqrt[3]{x y^{2}} \) was converted to involve only positive exponents: \( x^{\frac{1}{3}} y^{\frac{2}{3}} \).

Working with positive exponents helps to:

• Simplify expressions and mathematical problems.
• Facilitate easier application of exponent laws.
• Ensure coherence in equations and algebraic identities.

Having expressions with positive exponents makes evaluating and manipulating algebraic and numerical problems much simpler.