Problem 70
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. $$\frac{(5 x+3)^{1 / 2}-\frac{5}{2} x(5 x+3)^{-1 / 2}}{5 x+3}$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x = -\frac{6}{5}\).
1Step 1: Simplifying the Expression
We start by rewriting the expression with positive exponents. The original expression is:\[ \frac{(5x+3)^{1/2} - \frac{5}{2}x(5x+3)^{-1/2}}{5x+3} \]Rewrite the term \((5x+3)^{-1/2}\) using positive exponents:\[ (5x+3)^{-1/2} = \frac{1}{(5x+3)^{1/2}} \]Substitute back into the expression:\[ \frac{(5x+3)^{1/2} - \frac{5x}{2(5x+3)^{1/2}}}{5x+3} \]Combine the terms in the numerator:\[ \frac{(5x+3)^{1/2} - \frac{5x}{2(5x+3)^{1/2}}}{5x+3} = \frac{\frac{2(5x+3) - 5x}{2(5x+3)^{1/2}}}{5x+3} \]
2Step 2: Simplifying Further
Simplify the expression inside the numerator:\[ 2(5x+3) - 5x = 10x + 6 - 5x = 5x + 6 \]So the expression becomes:\[ \frac{\frac{5x + 6}{2(5x+3)^{1/2}}}{5x+3} \]Rewriting gives us:\[ \frac{5x + 6}{2(5x+3)^{3/2}} \]
3Step 3: Setting the Expression Equal to Zero
To find when the expression is zero, set the simplified expression equal to zero:\[ \frac{5x + 6}{2(5x+3)^{3/2}} = 0 \]Since a fraction is zero when its numerator is zero, set the numerator equal to zero:\[ 5x + 6 = 0 \]
4Step 4: Solve the Equation
Solve the linear equation:\[ 5x + 6 = 0 \]Subtract 6 from both sides:\[ 5x = -6 \]Divide by 5:\[ x = -\frac{6}{5} \]
Key Concepts
Simplifying ExpressionsNegative ExponentsSolving Linear Equations
Simplifying Expressions
Simplifying expressions can make them easier to understand and solve. It often involves combining like terms or removing complex parts of an expression. In our exercise, one key task was to change negative exponents to positive ones.
First, if you see an expression with a negative exponent, like \((5x+3)^{-1/2}\), you want to convert it into a positive exponent by using this rule: \(a^{-n} = \frac{1}{a^{n}}\). This changes \((5x+3)^{-1/2}\) to \(\frac{1}{(5x+3)^{1/2}}\). This step helps in simplifying the operation of division, as it's much easier to work with positive exponents.
Next, substitute back into the original expression and combine like terms where possible. For instance, in this exercise, we combined terms in the numerator to further simplify our expression. This makes it easier to see and understand, ultimately helping to solve problems like setting equations to 0.
First, if you see an expression with a negative exponent, like \((5x+3)^{-1/2}\), you want to convert it into a positive exponent by using this rule: \(a^{-n} = \frac{1}{a^{n}}\). This changes \((5x+3)^{-1/2}\) to \(\frac{1}{(5x+3)^{1/2}}\). This step helps in simplifying the operation of division, as it's much easier to work with positive exponents.
Next, substitute back into the original expression and combine like terms where possible. For instance, in this exercise, we combined terms in the numerator to further simplify our expression. This makes it easier to see and understand, ultimately helping to solve problems like setting equations to 0.
Negative Exponents
Negative exponents can at first seem a little daunting, but they’re actually quite straightforward once you know the key principle. The principle is that a negative exponent indicates division, rather than multiplication.
For example, \(x^{-1}\) is the same as \(\frac{1}{x}\). It's essentially telling us to "move" the base to the other side of the fraction, turning it into a positive exponent. A negative exponent doesn't mean a negative number, just a different form of the number.
In algebraic expressions, shifting negative exponents to the positive using the above transformation can greatly simplify things. We often shift all terms with negative exponents to make the entire expression work more smoothly and naturally. This step is crucial in problems like those involving fractions, where simplifying the equation is necessary to find the solution.
For example, \(x^{-1}\) is the same as \(\frac{1}{x}\). It's essentially telling us to "move" the base to the other side of the fraction, turning it into a positive exponent. A negative exponent doesn't mean a negative number, just a different form of the number.
In algebraic expressions, shifting negative exponents to the positive using the above transformation can greatly simplify things. We often shift all terms with negative exponents to make the entire expression work more smoothly and naturally. This step is crucial in problems like those involving fractions, where simplifying the equation is necessary to find the solution.
Solving Linear Equations
Once an expression is simplified, you often need to set it equal to zero and solve it. Solving linear equations involves finding the value for \(x\) that makes the equation true. In our exercise, once simplified, the expression had us working with a linear equation: \(5x + 6 = 0\).
This involves a couple of straightforward steps:
These steps follow basic arithmetic operations and logic, and they are critical skills in algebra. Solving linear equations like this is fundamental, allowing us to determine unknown values in a variety of mathematical and real-world problems.
This involves a couple of straightforward steps:
- First, isolate the term with the variable \(x\) by subtracting 6 from both sides, resulting in \(5x = -6\).
- Then, divide both sides by 5 to solve for \(x\), giving \(x = -\frac{6}{5}\).
These steps follow basic arithmetic operations and logic, and they are critical skills in algebra. Solving linear equations like this is fundamental, allowing us to determine unknown values in a variety of mathematical and real-world problems.
Other exercises in this chapter
Problem 69
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Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{1}{x^{2}+3}$$
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Wing Size (See Exercise 61.) Heavier birds have larger wings with more surface area than do lighter birds. For some species of birds, this relationship is given
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