Problem 73 - Intermediate Algebra

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{x^{3}} $$

Step-by-Step Solution

Answer

\( \sqrt[6]{x^3} = x^{1/2} \).

Step 1: Understand the Radical

The given expression is \( \sqrt[6]{x^3} \), which is a 6th root of \( x^3 \). To simplify this using rational exponents, we need to rewrite the radical in terms of exponents.

Step 2: Rewrite Radical as Rational Exponent

A radical of the form \( \sqrt[n]{x^m} \) can be written as \( x^{m/n} \). In this case, rewrite \( \sqrt[6]{x^3} \) as \( x^{3/6} \).

Step 3: Simplify the Rational Exponent

Simplify the fraction \( \frac{3}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This results in \( \frac{3}{6} = \frac{1}{2} \).

Step 4: Express the Simplified Exponent

The expression \( x^{3/6} \) simplifies to \( x^{1/2} \). Therefore, \( \sqrt[6]{x^3} = x^{1/2} \).

Key Concepts

Radical Simplification

Radical simplification involves taking expressions with roots and making them as simple as possible. Imagine a cluttered closet; what we're doing is organizing it. When dealing with radicals, like \( \sqrt[6]{x^3} \), we aim to express it in a more straightforward form using rational exponents.
To simplify a radical, start by recognizing the type of root we're dealing with. The expression \( \sqrt[6]{x^3} \) tells us it's a sixth root, making this a higher-index radical. Simplifying radicals often means converting them into an expression with fraction exponents, which can be easier to handle.

• Identifying the radical as a fraction helps in further simplification.
• Radical expressions often benefit from expressing them in relation to their simplest fractional counterparts.
• Using rational exponents instead of radicals allows easier manipulation and understanding of the expression.

By simplifying radicals like this, we make calculation and understanding much simpler and more approachable.

Fraction Exponents

Fraction exponents, or rational exponents, are a way to express roots and powers simultaneously in a very concise form. Rational exponents connect directly with roots through a neat mathematical tool: they are fractions where the numerator tells you the power and the denominator tells you the root. For instance, the fractional exponent in \( x^{3/6} \) implies taking \( x \) to the power of three, then finding the sixth root: \( \sqrt[6]{x^3} \).
In our solution, simplifying the fraction \( \frac{3}{6} \) to \( \frac{1}{2} \) means we're expressing \( x^{1/2} \). This illustrates a specific power of \( x \) at a glance.

• The numerator of the fraction is the power.
• The denominator indicates the root.

Fraction exponents let us shift seamlessly between powers and roots, making expressions much easier to manipulate. When simplified, they reveal that deeper connections lie between different mathematical operations.

Roots and Radicals

Roots and radicals are essential parts of algebra that describe taking a number and figuring out what value, multiplied by itself a certain number of times, gets you back to your original number. The simplest way of understanding radicals is to look at examples:
- \( \sqrt{x} \) is the square root because it pulls the number you would square to get \( x \).
- \( \sqrt[3]{x} \) is the cube root because cubing that number gives you back \( x \).
These operations involve finding a balance, returning to the foundational idea that roots point to something deeper about numbers.
For the problem at hand, the 6th root in \( \sqrt[6]{x^3} \) asks, "What multiplied six times gives me \( x^3 \)?".

• Understanding roots helps in breaking down complex algebraic expressions.
• Radicals show themselves in various forms; realizing this assists in identifying solutions quickly and efficiently.

Knowing your way around roots and radicals is essential for tackling advanced algebraic problems, opening the door to mastering these concepts.