Problem 107 - Algebra and Trigonometry

Question

Simplify by reducing the index of the radical. $$ \sqrt[9]{x^{6} y^{3}} $$

Step-by-Step Solution

Answer

\(x^{2/3} \cdot y^{1/3} \)

Step 1: Understand the Expressions

Given an expression \( \sqrt[9]{x^{6} y^{3}} \). Here, the expression inside the cube root is raised to the power 6 for \(x\) and to the power 3 for \(y\) . The index of the root is 9.

Step 2: Apply Exponent Rules

According to the rules of exponents, the expression \( \sqrt[9]{x^{6}} \) becomes \( {x^{6/9}} \) and \( \sqrt[9]{y^{3}} \) becomes \( {y^{3/9}} \).

Step 3: Simplify the Exponents

On simplifying the exponents, we get \(x^{6/9} = x^{2/3}\) and \(y^{3/9} = y^{1/3}\) .

Key Concepts

Exponent Rules

Exponent rules are essential tools in simplifying and manipulating expressions involving powers. They provide a way to handle calculations involving exponents in a straightforward manner. Here are some key exponent rules to remember:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, for bases like \(a^m \cdot a^n\), it becomes \(a^{m+n}\).
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. This transforms expressions like \(\frac{b^m}{b^n}\) into \(b^{m-n}\).
• Power of a Power Rule: When exponentiating an exponentiated base, multiply the exponents. For instance, \((a^m)^n\) becomes \(a^{mn}\).
• Power of a Product Rule: For a product raised to an exponent, apply the exponent to each factor separately: \((ab)^m = a^m b^m\).

Understanding these rules allows you to rewrite and simplify expressions systematically, making complex computations much easier.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. The expression \( \sqrt[n]{a} \) represents the \(n\)-th root of \(a\). Simplifying radical expressions often involves rewriting them in terms of exponents to make calculations more manageable.

In the original exercise, we tackle the expression \( \sqrt[9]{x^{6} y^{3}} \). By rewriting the radicals using exponent rules, we aim to simplify the expression. This involves expressing the roots as fractional exponents. The \(n\)-th root of a term is equivalent to raising the term to the power of \(1/n\). This rewiring changes \( \sqrt[9]{x^{6}} \) to \(x^{6/9}\) and \( \sqrt[9]{y^{3}} \) to \(y^{3/9}\).

Converting radicals to fractional exponents aids in recognizing patterns and relationships, setting the stage for further simplification.

Simplifying Exponents

Once radical expressions are converted to exponents, the next step is simplifying those exponents. This process involves reducing the fractions to their simplest form, which makes expressions clearer and easier to work with.

In our example, we take the exponent \( \frac{6}{9} \) for \(x\) and simplify it. Since both numbers, 6 and 9, are divisible by 3, we reduce \( \frac{6}{9} \) to \( \frac{2}{3} \). Similarly, for \(y\), we simplify \( \frac{3}{9} \) to \( \frac{1}{3} \).

This gives us the simplified expression \(x^{2/3}y^{1/3}\). This process simplifies computations and better reveals the structure of the equation or formula at play. Simplifying expressions using these methods can help solve many math problems more efficiently, making the process less daunting.