Problem 28 - College Algebra with Modeling and Visualization

Question

Use positive exponents to rewrite. $$ \frac{\sqrt[3]{x}}{\sqrt{x}} $$

Step-by-Step Solution

Answer

The expression is \(\frac{1}{x^{1/6}}\).

Step 1: Express Radicals as Exponents

First, express the roots in the expression as exponents. The cube root of \(x\) can be expressed as \(x^{1/3}\) and the square root of \(x\) as \(x^{1/2}\). Putting these into the fraction gives \(\frac{x^{1/3}}{x^{1/2}}\).

Step 2: Apply the Law of Exponents

Next, apply the law of exponents which states \(\frac{a^m}{a^n} = a^{m-n}\). Apply this to \(\frac{x^{1/3}}{x^{1/2}}\), resulting in \(x^{1/3 - 1/2}\).

Step 3: Subtract the Exponents

Now subtract the exponents: \(\frac{1}{3} - \frac{1}{2} = \frac{2}{6} - \frac{3}{6} = -\frac{1}{6}\). So, \(x^{1/3 - 1/2} = x^{-1/6}\).

Step 4: Rewrite Using Positive Exponent

Convert the negative exponent into a positive one using the rule \(a^{-n} = \frac{1}{a^n}\). Thus, \(x^{-1/6} = \frac{1}{x^{1/6}}\).

Key Concepts

Radicals as Exponents

Radicals are often seen in the form of square roots, cube roots, or any higher-order roots. To handle these easily in mathematical expressions, we can convert them into exponents.
For example, the cube root of a number is expressed as raising it to the power of one-third. Similarly, a square root translates into raising the number to one-half.

This conversion is particularly helpful when performing calculations because it streamlines operations that involve multiplying or dividing these roots. So, you could rewrite \( \sqrt[3]{x} \) as \( x^{1/3} \) and \( \sqrt{x} \) as \( x^{1/2} \). This is a fundamental step in simplifying many expressions.

Law of Exponents

The laws of exponents are a set of rules that simplify working with expressions that have exponents. Knowing these laws can simplify operations like multiplication or division.

One key law is the Quotient Rule. It says that when you divide two powers with the same base, you can subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \). This allows us to simplify expressions quickly.

• For example, applying this law to \( \frac{x^{1/3}}{x^{1/2}} \), we subtract the exponents to get \( x^{1/3 - 1/2} \).

Understanding how these rules work lays the groundwork for simplifying and solving complex algebraic expressions.

Negative Exponents

Negative exponents can seem intimidating at first, but they are very straightforward once you grasp their meaning. A negative exponent indicates that you should take the reciprocal of the base raised to the positive of that exponent.

In simpler terms, \( a^{-n} \) means \( \frac{1}{a^n} \). This concept comes in handy when you need to represent an expression with positive exponents.

• When you encounter an expression like \( x^{-1/6} \), you can rewrite this as \( \frac{1}{x^{1/6}} \).

This understanding helps in ensuring that all expressions are presented in positive exponent form, which is often preferred in mathematical work.

Simplifying Expressions

Simplifying expressions is about creating a form of the expression that is easier to understand and work with. This often involves using a combination of converting radicals, applying laws of exponents, and managing negative exponents.

Starting by converting all radicals into exponents makes exponent rules applicable.

• Next, apply the law of exponents to combine or reduce these expressions.
• Finally, convert any negative exponents into positive form for the simplest expression.

For example, simplifying \( \frac{\sqrt[3]{x}}{\sqrt{x}} \) led us to \( \frac{1}{x^{1/6}} \) after converting the radicals, subtracting the exponents, and addressing the negative exponent. This new form is more straightforward and functional for further algebraic processing.