Problem 54
Question
\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ \frac{s^{5 / 2}\left(2 s^{5 / 4}\right)^{2}}{s^{1 / 2}} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \\(4s^{9/2}\\).
1Step 1: Expand the Expression
Start by expanding the expression \(\left(2 s^{5/4}\right)^{2}\). By using the power of a product rule, \(\left(ab\right)^n = a^n b^n\), we get: \[\left(2 s^{5/4}\right)^{2} = 2^2 \cdot \left(s^{5/4}\right)^{2} = 4 \cdot s^{5/2}.\]
2Step 2: Substitute the Expansion
Substitute \(4 \cdot s^{5/2}\) back into the original expression: \[\frac{s^{5/2} \cdot 4s^{5/2}}{s^{1/2}}.\] This simplifies the expression to \[\frac{4s^{5/2 + 5/2}}{s^{1/2}} = \frac{4s^{10/2}}{s^{1/2}}.\]
3Step 3: Simplify the Rational Expression
Now, simplify \(\frac{4s^{10/2}}{s^{1/2}}\). First, simplify the exponents by subtracting them: \[\frac{4s^5}{s^{1/2}} = 4s^{5 - 1/2} = 4s^{10/2 - 1/2} = 4s^{9/2}.\] Therefore, the expression simplifies to \(4s^{9/2}\).
Key Concepts
ExponentsPower of a Product RuleRational ExpressionsSimplifying Expressions
Exponents
Exponents are a shorthand way of expressing repeated multiplication. When you see a number or variable raised to a power, it means to multiply that base by itself a certain number of times. For instance, when you encounter something like \( s^n \), the number \( n \) is the exponent and \( s \) is the base.
In our example, \( s^{5/2} \) means \( s \) is multiplied by itself \( 2.5 \) times, which can also be expressed as the square root of \( s \) taken five times. Understanding exponents is crucial because it underpins many simplification steps in algebra such as combining like terms and expanding expressions.
In our example, \( s^{5/2} \) means \( s \) is multiplied by itself \( 2.5 \) times, which can also be expressed as the square root of \( s \) taken five times. Understanding exponents is crucial because it underpins many simplification steps in algebra such as combining like terms and expanding expressions.
Power of a Product Rule
The power of a product rule helps us simplify expressions where a product of terms is raised to a power. According to this rule, \((ab)^n = a^n \cdot b^n\). This means you can distribute the exponent to each term inside the parentheses.
In the exercise, you have \((2s^{5/4})^2\). Applying the power of a product rule, we break this down into \(2^2 \cdot (s^{5/4})^2 = 4 \cdot s^{5/2}\). Using this rule simplifies the expression and sets the stage for further simplifications.
By applying rules like this, we focus on each part of the expression separately and manage complex calculations more effectively.
In the exercise, you have \((2s^{5/4})^2\). Applying the power of a product rule, we break this down into \(2^2 \cdot (s^{5/4})^2 = 4 \cdot s^{5/2}\). Using this rule simplifies the expression and sets the stage for further simplifications.
By applying rules like this, we focus on each part of the expression separately and manage complex calculations more effectively.
Rational Expressions
Rational expressions are like fractions but instead they involve polynomials in the numerator and/or denominator. Just like regular fractions, they can be simplified and manipulated by finding common factors or using algebraic rules.
In our solution, we deal with a rational expression \(\frac{s^{5/2} \cdot 4s^{5/2}}{s^{1/2}}\). Our goal is to reduce this expression by canceling terms and applying exponent rules. Simplifying rational expressions can make it easier to solve problems or understand the relationships between different variables.
Understanding rational expressions is a valuable skill in algebra, calculus, and many real-world applications.
In our solution, we deal with a rational expression \(\frac{s^{5/2} \cdot 4s^{5/2}}{s^{1/2}}\). Our goal is to reduce this expression by canceling terms and applying exponent rules. Simplifying rational expressions can make it easier to solve problems or understand the relationships between different variables.
Understanding rational expressions is a valuable skill in algebra, calculus, and many real-world applications.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This usually means performing operations, combining like terms, and using algebraic rules to eliminate unnecessary parts.
In this problem, after expanding with the power of a product rule and substituting back, we simplify the expression \(\frac{4s^{10/2}}{s^{1/2}}\). By subtracting the exponents, we achieve \(4s^{9/2}\), which is the simplest form of the original expression.
Simplifying expressions makes your answers more readable and often sets the stage for further mathematical manipulations. It also helps ensure accuracy when working functionally with multiple equations.
In this problem, after expanding with the power of a product rule and substituting back, we simplify the expression \(\frac{4s^{10/2}}{s^{1/2}}\). By subtracting the exponents, we achieve \(4s^{9/2}\), which is the simplest form of the original expression.
Simplifying expressions makes your answers more readable and often sets the stage for further mathematical manipulations. It also helps ensure accuracy when working functionally with multiple equations.
Other exercises in this chapter
Problem 54
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Perform the addition or subtraction and simplify. $$ \frac{1}{x^{2}}+\frac{1}{x^{2}+x} $$
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