Problem 93 - Algebra and Trigonometry

Question

Simplify using properties of exponents. $$ \frac{20 x^{\frac{1}{2}}}{5 x^{\frac{1}{4}}} $$

Step-by-Step Solution

Answer

The simplified version of the expression \( \frac{20 x^{\frac{1}{2}}}{5 x^{\frac{1}{4}}} \) is \( 4x^{1/4} \)

Step 1: Simplify the Coefficients

First, simplify the coefficients by dividing 20 by 5, which will give us 4.

Step 2: Apply the Division Rule for Exponents

Next, apply the exponent rule for division to the terms with the same base, \(x\). To do this, you subtract the exponent in the denominator from the exponent in the numerator. For this case, \( x^{1/2}/x^{1/4} = x^{1/2 - 1/4} \).

Step 3: Subtract the Exponents

Subtract the exponents \(1/2\) and \(1/4\). After doing this, you’ll get \(1/4\) as a result. So, \( x^{1/2 - 1/4} = x^{1/4} \).

Step 4: Write the Final Answer

Finally, write down the remaining coefficient from the first step together with the result from the prior step. This gives us the final simplified answer: \( 4x^{1/4} \)

Key Concepts

Simplifying Expressions

When faced with a complicated expression, the goal is to make it as simple as possible. This involves reducing the expression to its simplest form without changing its value. Simplifying expressions often involves several steps, such as combining like terms, reducing fractions, and applying mathematical rules.
In the context of the original exercise, simplifying the expression involves both simplifying the numerical coefficients and managing the exponents. Break down the problem step by step:

• First, simplify the coefficients. This means dividing the numbers in front of the exponents. Here, 20 is divided by 5 to obtain 4.
• Next, handle the exponents which typically involves applying properties or rules of exponents. This will make the entire expression easier to understand and work with.

When you simplify expressions this way, it becomes much clearer and allows for easier computation in further mathematical operations.

Division Rule for Exponents

The division rule for exponents is a crucial property to understand when simplifying expressions. If you have the same base and are dividing two exponential expressions, you can simplify by subtracting the exponent in the denominator from the exponent in the numerator.
Mathematically, this rule states:

• \( \frac{a^m}{a^n} = a^{m-n} \)

Here, "a" represents the base, and "m" and "n" are the exponents. It tells us how to combine two powers with the same base during division.
In the given problem, we apply this rule to the variable \(x\):

• \( \frac{x^{1/2}}{x^{1/4}} = x^{1/2 - 1/4} \)

This subtraction simplifies the division's complexity, leaving a clearer, more manageable expression.

Subtraction of Exponents

When dealing with exponential expressions that need to be simplified, subtraction of exponents is often used, especially in division. This is straightforward when both exponents are fractions.
In the exercise example:

• The problem requires us to subtract the exponent \(\frac{1}{4}\) from \(\frac{1}{2}\).
• Using simple fraction subtraction, \(\frac{1}{2}\) converts to \(\frac{2}{4}\), making it easy to subtract: \(\frac{2}{4} - \frac{1}{4} = \frac{1}{4}\).

By subtracting the exponents, the expression simplifies:

• \(x^{1/2 - 1/4} = x^{1/4}\)

This results in a simpler form that's easier to interpret and further work with.

Rational Exponents

Rational exponents, simply put, are exponents that are fractions. They offer an alternative way to represent roots, as well as perform operations like multiplication and division with exponents more flexibly.
Consider an expression with a rational exponent:

• \(x^{\frac{m}{n}}\) is equivalent to \(\sqrt[n]{x^m}\) or \((\sqrt[n]{x})^m\).

In the original example, the expression \(x^{\frac{1}{2}}\) is similar to the square root of \(x\), and \(x^{\frac{1}{4}}\) equates to the fourth root of \(x\).
Rational exponents allow us to perform mathematical operations easily, as seen in the original exercise, where calculating with these terms leads to simplified forms like \(4x^{\frac{1}{4}}\). Understanding how to manipulate these exponents paves the way for elegant solutions and deeper comprehension of exponential properties.