Problem 67
Question
Complete the following. (a) Simplify the given expression so that it does not have negative exponents. (b) Set the expression from part (a) equal to 0 and solve the resulting equation. $$\frac{x^{1 / 4}-x^{-3 / 4}}{x}$$
Step-by-Step Solution
Verified Answer
The solution is \( x = 1 \).
1Step 1: Simplify the Expression
The initial expression is \( \frac{x^{1/4} - x^{-3/4}}{x} \). We start by expressing \( x \) in the denominator as a power of \( x \), which is \( x^1 \). We can rewrite the expression as \( \frac{x^{1/4} - x^{-3/4}}{x^1} \). Now, apply the rule of exponents \( \frac{a^m}{a^n} = a^{m-n} \). This results in, \( x^{1/4 - 1} - x^{-3/4 - 1}\), which simplifies to \( x^{-3/4} - x^{-7/4} \). To remove negative exponents, each term can be rewritten as \( \frac{1}{x^{3/4}} - \frac{1}{x^{7/4}} \). Finally, express it as a single fraction: \( \frac{x^{7/4} - x^{3/4}}{x^{7/4}} \).
2Step 2: Set Expression Equal to Zero
Now, set the simplified expression from Step 1 equal to zero: \( \frac{x^{7/4} - x^{3/4}}{x^{7/4}} = 0 \). A fraction equals zero when its numerator equals zero, so we solve the equation \( x^{7/4} - x^{3/4} = 0 \).
3Step 3: Solve the Equation
Factor out \( x^{3/4} \) from the equation, resulting in \( x^{3/4}(x^{1} - 1) = 0 \). This gives us two solutions: \( x^{3/4} = 0 \) and \( x^{1} - 1 = 0 \). Since \( x^{3/4} = 0 \) implies \( x = 0 \), we consider this a potential solution. The second equation, \( x^{1} - 1 = 0 \), simplifies to \( x = 1 \). But \( x = 0 \) is not valid because of the division by zero in the original expression. Therefore, the solution is \( x = 1 \).
Key Concepts
Simplifying ExpressionsNegative ExponentsSolving Equations
Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra that helps in making complex equations more manageable. For this specific exercise, we need to simplify the expression \( \frac{x^{1/4} - x^{-3/4}}{x} \).
The first step is to handle the denominator, \( x \), so it matches the exponents in the numerator. We rewrite it as \( x^1 \).
Using the properties of exponents, specifically \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify each term in the numerator. For example, \( x^{1/4} \div x^1 = x^{1/4 - 1} \), resulting in \( x^{-3/4} \). Likewise, \( x^{-3/4} \div x^1 = x^{-3/4 - 1} = x^{-7/4} \).
This simplification gives us: \(- x^{7/4} \). This expression is simpler but still contains negative exponents, which we will handle next.
The first step is to handle the denominator, \( x \), so it matches the exponents in the numerator. We rewrite it as \( x^1 \).
Using the properties of exponents, specifically \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify each term in the numerator. For example, \( x^{1/4} \div x^1 = x^{1/4 - 1} \), resulting in \( x^{-3/4} \). Likewise, \( x^{-3/4} \div x^1 = x^{-3/4 - 1} = x^{-7/4} \).
This simplification gives us: \(- x^{7/4} \). This expression is simpler but still contains negative exponents, which we will handle next.
Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponents. Converting expressions with negative exponents into fractions with positive exponents is essential for further simplification.
In our expression \( x^{-3/4} - x^{-7/4} \), convert negative exponents like this: \( x^{-3/4} = \frac{1}{x^{3/4}} \) and \( x^{-7/4} = \frac{1}{x^{7/4}} \).
The expression becomes \( \frac{1}{x^{3/4}} - \frac{1}{x^{7/4}} \).
To express it as a single fraction, find a common denominator, here \( x^{7/4} \). This yields \( \frac{x^{7/4} - x^{3/4}}{x^{7/4}} \).
Working with negative exponents in this way gives clarity and order, preparing for solving the equation effectively.
In our expression \( x^{-3/4} - x^{-7/4} \), convert negative exponents like this: \( x^{-3/4} = \frac{1}{x^{3/4}} \) and \( x^{-7/4} = \frac{1}{x^{7/4}} \).
The expression becomes \( \frac{1}{x^{3/4}} - \frac{1}{x^{7/4}} \).
To express it as a single fraction, find a common denominator, here \( x^{7/4} \). This yields \( \frac{x^{7/4} - x^{3/4}}{x^{7/4}} \).
Working with negative exponents in this way gives clarity and order, preparing for solving the equation effectively.
Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. At this stage, our simplified expression \( \frac{x^{7/4} - x^{3/4}}{x^{7/4}} \) is set to zero, meaning \( x^{7/4} - x^{3/4} = 0 \).
A fraction equals zero when its numerator is zero, so solve by factoring the common term from the numerator: \( x^{3/4} \).
This results in \( x^{3/4} (x^{1} - 1) = 0 \), leading to two equations: \( x^{3/4} = 0\) and \( x^{1} - 1 = 0 \).
For \( x^{1} - 1 = 0 \), it simplifies to \( x = 1 \), a valid solution since it doesn't result in division by zero in the original expression. However, \( x^{3/4} = 0 \) implies \( x = 0 \), but this creates a division by zero in the original equation, invalidating it as a solution.
Thus, the valid solution for the equation is \( x = 1 \). By understanding these steps, you can confidently solve similar algebraic equations.
A fraction equals zero when its numerator is zero, so solve by factoring the common term from the numerator: \( x^{3/4} \).
This results in \( x^{3/4} (x^{1} - 1) = 0 \), leading to two equations: \( x^{3/4} = 0\) and \( x^{1} - 1 = 0 \).
For \( x^{1} - 1 = 0 \), it simplifies to \( x = 1 \), a valid solution since it doesn't result in division by zero in the original expression. However, \( x^{3/4} = 0 \) implies \( x = 0 \), but this creates a division by zero in the original equation, invalidating it as a solution.
Thus, the valid solution for the equation is \( x = 1 \). By understanding these steps, you can confidently solve similar algebraic equations.
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