Problem 594
Question
In the following exercises, simplify. $$ \frac{w^{\frac{2}{5}}}{\frac{7}{5}} $$
Step-by-Step Solution
Verified Answer
The simplified form is \( w^{\frac{2}{5}} \).
1Step 1 - Understanding the Expression
First, identify the given expression: \[ \frac{w^{\frac{2}{5}}}{\frac{7}{5}} \]. This is a fraction where the numerator is a term with a variable and the denominator is a constant.
2Step 2 - Simplify the Denominator
Notice that the denominator is \( \frac{7}{5} \). To simplify the entire fraction, we can multiply both the numerator and the denominator by the reciprocal of the denominator to eliminate it. The reciprocal of \( \frac{7}{5} \) is \( \frac{5}{7} \). So, multiply both the numerator and the denominator by \( \frac{5}{7} \): \[ \frac{w^{\frac{2}{5}}}{\frac{7}{5}} \times \frac{5}{7} = w^{\frac{2}{5}} \times \frac{5}{7} \times \frac{7}{5} \].
3Step 3 - Simplify the Multiplication
The terms \( \frac{5}{7} \) and \( \frac{7}{5} \) cancel each other out: \[ w^{\frac{2}{5}} \times \frac{5}{7} \times \frac{7}{5} = w^{\frac{2}{5}} \times 1 \].
4Step 4 - Write the Final Answer
The expression simplifies directly to \( w^{\frac{2}{5}} \).
Key Concepts
Fraction SimplificationReciprocalAlgebraic ExpressionsExponents
Fraction Simplification
Simplifying a fraction means to make it easier to understand and work with. In algebra, a fraction can include variables, constants, or a mix of both.
To simplify a fraction, we often try to reduce it to its simplest form, where the greatest common factor in the numerator and the denominator is 1.
For example, in the exercise given, the fraction \(\frac{w^{\frac{2}{5}}}{\frac{7}{5}}\) can be simplified by eliminating the denominator through multiplication with its reciprocal.
When you perform these steps, the goal is to achieve a fraction that is more straightforward, so all unnecessary complexity is removed.
To simplify a fraction, we often try to reduce it to its simplest form, where the greatest common factor in the numerator and the denominator is 1.
For example, in the exercise given, the fraction \(\frac{w^{\frac{2}{5}}}{\frac{7}{5}}\) can be simplified by eliminating the denominator through multiplication with its reciprocal.
When you perform these steps, the goal is to achieve a fraction that is more straightforward, so all unnecessary complexity is removed.
Reciprocal
A reciprocal of a number or fraction is what you multiply it by to get a product of 1.
For any fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
To find the reciprocal, simply switch the numerator and the denominator.
In the context of our exercise, to simplify \(\frac{w^{\frac{2}{5}}}{\frac{7}{5}}\), we used the reciprocal of \(\frac{7}{5}\), which is \(\frac{5}{7}\). By multiplying both the numerator and the denominator of our original fraction by the reciprocal, we successfully eliminate the fractional denominator, making the fraction easier to handle.
For any fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
To find the reciprocal, simply switch the numerator and the denominator.
In the context of our exercise, to simplify \(\frac{w^{\frac{2}{5}}}{\frac{7}{5}}\), we used the reciprocal of \(\frac{7}{5}\), which is \(\frac{5}{7}\). By multiplying both the numerator and the denominator of our original fraction by the reciprocal, we successfully eliminate the fractional denominator, making the fraction easier to handle.
Algebraic Expressions
An algebraic expression includes numbers, variables, and operational signs.
They can represent single values or complex relationships between numbers and variables.
In the given exercise, \(\frac{w^{\frac{2}{5}}}{\frac{7}{5}}\) is an algebraic fraction.
By simplifying this expression, we learn to manage expressions where variables appear in both the numerator and the denominator.
Simplifying these types of algebraic fractions is key to solving more complex algebraic equations and understanding higher math concepts.
They can represent single values or complex relationships between numbers and variables.
In the given exercise, \(\frac{w^{\frac{2}{5}}}{\frac{7}{5}}\) is an algebraic fraction.
By simplifying this expression, we learn to manage expressions where variables appear in both the numerator and the denominator.
Simplifying these types of algebraic fractions is key to solving more complex algebraic equations and understanding higher math concepts.
Exponents
Exponents show how many times a number (called the base) is multiplied by itself.
For example, \(a^n\) means the base \(a\) is used as a factor \(n\) times.
In the exercise provided, \(w^{\frac{2}{5}}\) is an example of a fractional exponent, which some might find intimidating.
However, it simply means that \(w\) is raised to the power of \(\frac{2}{5}\).
Simplifying expressions with fractional exponents often involves understanding and applying the rules of exponents for combining, distributing, and simplifying terms.
In this case, we used these rules to simplify our algebraic fraction to its cleanest form.
For example, \(a^n\) means the base \(a\) is used as a factor \(n\) times.
In the exercise provided, \(w^{\frac{2}{5}}\) is an example of a fractional exponent, which some might find intimidating.
However, it simply means that \(w\) is raised to the power of \(\frac{2}{5}\).
Simplifying expressions with fractional exponents often involves understanding and applying the rules of exponents for combining, distributing, and simplifying terms.
In this case, we used these rules to simplify our algebraic fraction to its cleanest form.
Other exercises in this chapter
Problem 592
In the following exercises, simplify. $$ \left(a^{24}\right)^{\frac{1}{6}} $$
View solution Problem 593
In the following exercises, simplify. $$ \left(b^{10}\right)^{\frac{3}{5}} $$
View solution Problem 595
In the following exercises, simplify. $$ \frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}} $$
View solution Problem 596
In the following exercises, simplify. $$ \left(27 r^{\frac{3}{5}}\right)^{\frac{1}{3}} $$
View solution