Problem 75 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \frac{\sqrt[4]{x^{7}}}{\sqrt[4]{x^{3}}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x \).

Step 1: Express using Rational Exponents

The expression given is \( \frac{\sqrt[4]{x^7}}{\sqrt[4]{x^3}} \). We can express the roots as rational exponents: \( \sqrt[4]{x^7} = x^{7/4} \) and \( \sqrt[4]{x^3} = x^{3/4} \). This transforms the expression into \( \frac{x^{7/4}}{x^{3/4}} \).

Step 2: Use the Quotient Rule for Exponents

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). Applying this to our expression, \( \frac{x^{7/4}}{x^{3/4}} = x^{7/4 - 3/4} \).

Step 3: Simplify the Exponents

Subtract the exponents: \( 7/4 - 3/4 = 4/4 = 1 \). Thus, the expression simplifies to \( x^1 \).

Step 4: Final Simplification

Since any number to the power of 1 is itself, \( x^1 = x \). Therefore, the simplified expression is \( x \).

Key Concepts

Quotient Rule for Exponents

The quotient rule for exponents is a crucial concept when working with algebraic expressions involving powers. This rule provides a way to divide expressions with the same base by simply subtracting the exponents.
In general terms, this rule is presented as \( \frac{a^m}{a^n} = a^{m-n} \). What it suggests is that when you have two exponents with the same base being divided, you can subtract the exponent in the denominator from the exponent in the numerator.
For example, in our exercise, the expression \( \frac{x^{7/4}}{x^{3/4}} \) uses the quotient rule by subtracting the exponent \( 3/4 \) from \( 7/4 \). This results in \( x^{7/4 - 3/4} = x^{4/4} \), which further simplifies to \( x^1 \).
This rule not only makes calculations more straightforward but also simplifies complex expressions, making them easier to work with in further algebraic manipulations.

Simplifying Expressions

Simplifying expressions in algebra is about reducing an equation or formula to its simplest form. This does not mean changing its value; rather, it allows for clearer understanding and more efficient computations.
Simplification involves combining like terms, reducing fractions, and applying mathematical rules such as the quotient rule.
In the exercise, we start by converting the roots into rational exponents: \( \sqrt[4]{x^7} = x^{7/4} \) and \( \sqrt[4]{x^3} = x^{3/4} \). This step is crucial as it sets up the expression for further simplification using exponent rules.
The primary goal is to reach a version of the expression that is concise and easy to interpret. By applying the quotient rule, the expression simplifies to \( x \), essentially showing that the original expression had unnecessary complexity for practical uses.

Algebraic Simplification

Algebraic simplification encompasses the techniques used to reduce complex algebraic expressions into simpler, more manageable forms.
It involves a clear understanding of algebraic rules and properties to reorganize and reduce expressions without changing their actual value.
In our example, algebraic simplification begins with the translation of roots into rational exponents, \( x^{7/4} \) and \( x^{3/4} \).
Once in rational form, the quotient rule guides the simplification process by subtracting exponents: \( 7/4 - 3/4 = 1 \). This transforms the expression to \( x^1 \), which further reduces to \( x \).
Algebraic simplification is essential in solving equations, allowing mathematicians and students to concentrate on the critical components of the expression without distractions from more complicated terms or forms.