Problem 98 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for the indicated variable. Assume all denominators are nonzero. $$x^{2 / 3}+y^{2 / 3}=a^{2 / 3}, \quad \text { for } y$$

Step-by-Step Solution

Answer

y = \left( a^{2/3} - x^{2/3} \right)^{3/2}

Step 1: Isolate the term involving y

Start by isolating the term that contains y on one side of the equation. Subtract \(x^{2/3}\) from both sides to get: \[ y^{2/3} = a^{2/3} - x^{2/3} \]

Step 2: Solve for y

To isolate y, raise both sides of the equation to the power of \frac{3}{2}\. This will eliminate the fractional exponent: \[ y = \left( a^{2/3} - x^{2/3} \right)^{3/2} \]

Key Concepts

Isolating Variables

Isolating a variable in an equation means rearranging the equation so that the variable you're solving for stands alone on one side of the equation. By doing this, you make it easier to understand the relationship between the variables.
To isolate a variable, you perform operations such as addition, subtraction, multiplication, and division to both sides of the equation until the variable is on its own.
In the given exercise, the goal is to solve for \(y\). Here’s a step-by-step breakdown:

• Start with the equation: \(x^{2/3} + y^{2/3} = a^{2/3}\).
• Subtract \(x^{2/3}\) from both sides: This gives \(y^{2/3} = a^{2/3} - x^{2/3}\). Now, the term involving \(y\) is isolated on one side of the equation.

Isolating variables is a fundamental skill in algebra, and it’s crucial for solving different kinds of equations.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. Understanding how to work with them is essential for solving equations like the one in this exercise.

• Fractional exponent definition: If you have an exponent of the form \(a^{m/n}\), it means you take the \(n\)-th root of \(a\) and then raise the result to the \(m\)-th power.
• Example: \(a^{1/2}\) is the same as \(\sqrt{a}\), and \(a^{3/2}\) is \(\left(\sqrt{a}\right)^3\).

In the exercise, the fractional exponent \(2/3\) will be undone by raising each side to the power of \(3/2\):

• Equation: \(y^{2/3} = a^{2/3} - x^{2/3}\)
• Raise both sides to the power of \(3/2\): This results in \(y = \left(a^{2/3} - x^{2/3}\right)^{3/2}\).

By raising both sides to the power of \(3/2\), the fractional exponent is neutralized, isolating \(y\).

Algebraic Manipulation

Algebraic manipulation involves using various algebraic techniques to rearrange an equation or expression into a more desirable form. This often includes using properties of algebraic operations, such as the distributive property and rules of exponents.
For the given problem, algebraic manipulation is crucial. Here’s the process:

• Identify the goal: The goal is to solve for \(y\).
• Isolate the term involving \(y\): Move terms around so the term with \(y\) is alone on one side of the equation.
• Deal with exponents: Use appropriate exponent rules to simplify the expression.

In the exercise, the steps are:

• Start with the given equation: \(x^{2/3} + y^{2/3} = a^{2/3}\)
• Isolate \(y^{2/3}\): Subtract \(x^{2/3}\) from both sides: \(y^{2/3} = a^{2/3} - x^{2/3}\)
• Solve for \(y\): Raise both sides to the power of \(3/2\): \(y = \left(a^{2/3} - x^{2/3}\right)^{3/2}\)

By understanding these algebraic techniques, you’ll be able to solve a wide range of equations efficiently and accurately.