Problem 106 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The reduction of the index of the radical \( \sqrt[9]{x^{6}} \) results in \(x^{\frac{2}{3}}\)

Step 1: Observe the given radical

The given radical is the 9th root of \(x^{6}\), written as \( \sqrt[9]{x^{6}} \)

Step 2: Apply the power rule for radicals

The power rule for radicals states that \(\sqrt[n]{a^{m}}\) can be simplified as \(a^{\frac{m}{n}}\). Hence, we can write \( \sqrt[9]{x^{6}} \) as \(x^{\frac{6}{9}}\)

Step 3: Simplify the fraction

The fraction \( \frac{6}{9} \) can be reduced to \(\frac{2}{3}\) by dividing both the numerator and denominator by their greatest common divisor, which is 3. Hence, the radical simplifies to \(x^{\frac{2}{3}}\)

Key Concepts

Understanding the Power Rule for Radicals

When you encounter a radical expression like \( \sqrt[n]{a^{m}} \), it might seem daunting at first. But the power rule for radicals makes it simpler.

This rule helps transform a radical expression into a more familiar format using exponents. The rule states that if you have a radical like \( \sqrt[n]{a^{m}} \), you can rewrite it as \( a^{\frac{m}{n}} \).

This transformation makes calculations easier, as you are now working with powers and fractions instead of roots. It's especially helpful for simplifying expressions, like in our exercise where \( \sqrt[9]{x^{6}} \) becomes \( x^{\frac{6}{9}} \).

Understanding this rule is crucial. It allows you to apply exponent rules effectively, making it a key step in simplifying radicals.

A Closer Look at Exponent Rules

Once you've used the power rule for radicals, you're left with an expression that involves exponents. It's important to brush up on your exponent rules to simplify these expressions further.

Here are some basic exponent rules you'll find useful:

• Product of Powers: When you multiply two powers with the same base, you add the exponents. For example, \( a^{m} \times a^{n} = a^{m+n} \).
• Quotient of Powers: When dividing two powers with the same base, subtract the exponents: \( \frac{a^{m}}{a^{n}} = a^{m-n} \).
• Power of a Power: To raise a power to another power, multiply the exponents: \( (a^{m})^{n} = a^{m \times n} \).
• Negative Exponents: A negative exponent indicates a reciprocal: \( a^{-n} = \frac{1}{a^{n}} \).
• Fractional Exponents: Fractional exponents denote roots: \( a^{\frac{1}{n}} = \sqrt[n]{a} \).

In our exercise, once \( x^{\frac{6}{9}} \) was obtained, simplifying the fraction was the next step, allowing us to employ the use of greatest common divisors.

Greatest Common Divisor and Simplification

The greatest common divisor (GCD) plays a significant role when dealing with fractions, particularly in simplifying them.

The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. To simplify a fraction, like \( \frac{6}{9} \), find the GCD of the numerator (6) and the denominator (9).

Steps to find the GCD:

• List the factors of each number. Factors of 6 are 1, 2, 3, 6. Factors of 9 are 1, 3, 9.
• Identify the greatest factor common to both lists, which is 3 in this case.

Use the GCD to divide both the numerator and the denominator:

• \( \frac{6}{3} = 2 \) and \( \frac{9}{3} = 3 \)
• The simplified fraction is \( \frac{2}{3} \)

This means the expression \( x^{\frac{6}{9}} \) becomes \( x^{\frac{2}{3}} \). Finding the GCD isn't just useful in math class—it's a handy skill that helps simplify problems in real life too.