Problem 95 - Elementary and Intermediate Algebra

Question

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[9]{11^{3}} $$

Step-by-Step Solution

Answer

The simplified form is \( 11^{\frac{1}{3}} \).

Step 1: Understand the Problem

We want to simplify the expression \( \sqrt[9]{11^{3}} \) using rational exponents. A rational exponent indicates both the root and the power of a number.

Step 2: Express the Radical as a Rational Exponent

The expression \( \sqrt[9]{11^{3}} \) can be rewritten using rational exponents as \( 11^{\frac{3}{9}} \). The denominator (9) in the rational exponent represents the 9th root, and the numerator (3) is the power.

Step 3: Simplify the Fraction

The fraction \( \frac{3}{9} \) can be simplified. Divide both the numerator and the denominator by their greatest common divisor, which is 3, to get \( \frac{1}{3} \).

Step 4: Rewrite the Expression with the Simplified Exponent

After simplifying the fraction, the expression \( 11^{\frac{3}{9}} \) becomes \( 11^{\frac{1}{3}} \).

Step 5: Interpret the Result

The expression \( 11^{\frac{1}{3}} \) corresponds to the cube root of 11. This is the simplified form of the original radical expression.

Key Concepts

Radical Simplification

The process of radical simplification is all about making expressions with roots more straightforward and concise. When you see a root, such as in the expression \( \sqrt[9]{11^{3}} \), it might look complex at first. However, we can simplify it using the concept of rational exponents. Here's the trick: a root can be expressed as an exponent. Specifically, the \( n \)-th root of a number is the same as raising that number to an exponent of \( \frac{1}{n} \).

• For instance, the 9th root \( \sqrt[9]{x} \) can be written as \( x^{\frac{1}{9}} \).

By understanding this conversion, radical expressions that look complicated can be rewritten in a simpler exponents form, allowing for easier manipulation and simplification.

Cube Root

The cube root is a common way to simplify expressions with rational exponents. If an expression is written as \( x^{\frac{1}{3}} \), it corresponds to the cube root of \( x \), which means finding a number that, when multiplied by itself twice, gives \( x \).

• For instance, if \( x = 8 \), then \( 8^{\frac{1}{3}} = 2 \) because \( 2 \times 2 \times 2 = 8 \).

The cube root is just a special case of the root, where the root index \( n \) is 3. The cube root appears frequently because it often creates a neat simplification of expressions.

Fraction Simplification

Simplifying fractions is a key step in cleaning up expressions involving rational exponents. Often, you will encounter fractions as exponents, like \( \frac{3}{9} \), which can usually be broken down into simpler forms.The rule of thumb is to reduce the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD).

• For example, in the fraction \( \frac{3}{9} \), the GCD is 3, so it can be simplified to \( \frac{1}{3} \).

Once simplified, the expression becomes easier to interpret; in this case, equating to the cube root of the base number. Understanding and applying fraction simplification is integral to working with rational exponents effectively.