Problem 74 - Intermediate Algebra

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[9]{a^{3}} $$

Step-by-Step Solution

Answer

The simplified form is \( a^{1/3} \).

Step 1: Convert Radical to Exponent

Recognize that the radical expression \( \sqrt[9]{a^3} \) can be expressed using rational exponents. The expression \( \sqrt[n]{x} \) is equivalent to \( x^{1/n} \). Therefore, \( \sqrt[9]{a^3} \) can be written as \( (a^3)^{1/9} \).

Step 2: Apply Power of a Power Property

Use the power of a power property, which states that \( (x^m)^n = x^{m \cdot n} \). So, apply this property to \( (a^3)^{1/9} \). Calculate the exponent by multiplying \( 3 \times \frac{1}{9} \), resulting in \( a^{3/9} \).

Step 3: Simplify the Exponent

Simplify \( \frac{3}{9} \) by dividing both the numerator and the denominator by the greatest common divisor, which is 3. This simplification gives us \( \frac{1}{3} \). Therefore, \( a^{3/9} \) simplifies to \( a^{1/3} \).

Key Concepts

Radicals

Radicals are mathematical symbols used to represent roots, commonly seen as square roots or cube roots. They are a way to express numbers that are not perfect squares or cubes, offering a method to interact with non-whole numbers in a manageable way. For instance, the radical expression \( \sqrt[9]{a^3} \) is read as the ninth root of \( a^3 \). The index, which is 9 in this case, tells us the degree of the root.
While radicals provide a familiar way to express roots, converting them to rational exponents can often simplify calculations. They help us to handle complex expressions, especially when working with multiple layers of roots and powers.
Understanding radicals is critical as they are foundational to exploring more advanced algebraic concepts, including those involving exponents and rational expressions.

Power of a Power Property

The power of a power property is a key algebraic rule that deals with exponents. This property states: \( (x^m)^n = x^{m \cdot n} \). This allows us to condense exponential expressions that involve powers raised to additional powers.
Take the expression \( (a^3)^{1/9} \) as an example. By applying the power of a power property, we can simplify to \( a^{3/9} \). Here, we multiplied the exponents—3 and \( \frac{1}{9} \)—to simplify the expression into a single power of \( a \).
This property enables us to simplify complicated exponential expressions by converting them to more straightforward formats, which is essential in solving higher-level equations and rationalizing complicated fractions.

Rational Expressions

Rational expressions are fractions that contain polynomials in their numerator and denominator. They are similar to numerical fractions but involve variables, offering a broader and more abstract application of the concepts learned with simple fractions.
In the context of our exercise, when we simplify a radical to an expression like \( a^{1/9} \), we are essentially dealing with a rational exponent, which is a type of rational expression. Rational exponents allow us to express radicals as fractions, making them easier to manipulate algebraically.
Rational expressions can seem complex, but the key is simplifying them to their most reduced form. This makes calculations easier and can reveal connections between variables that are not immediately obvious in the original expression.

Simplifying Exponents

Simplifying exponents is all about breaking down complex power expressions to their simplest form. This involves using rules like the power of a power property and knowledge of arithmetic operations.
For example, in our exercise, we converted \( \frac{3}{9} \) into \( \frac{1}{3} \) by identifying and applying the greatest common divisor. Simplified exponents like \( a^{1/3} \) are easier to work with and understand because they are in their most reduced format.
Simplifying keeps mathematical work clean, efficient, and more intuitive. This process reduces errors in calculation and enhances understanding, crucial for higher-level math courses that build on these foundational skills.