Solve each equation for the specified variable. (Assume all denominators are nonzero.) x^2/3+y^2/3=a^2/3, for y

Verified step by step guidance

1. Start with the given equation: \(x^{2\/3} + y^{2\/3} = a^{2\/3}\).
2. Isolate the term containing \(y\) by subtracting \(x^{2\/3}\) from both sides: \(y^{2\/3} = a^{2\/3} - x^{2\/3}\).
3. To solve for \(y\), raise both sides of the equation to the power that is the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\), to undo the fractional exponent on \(y\): \(\left(y^{2\/3}\right)^{3\/2} = \left(a^{2\/3} - x^{2\/3}\right)^{3\/2}\).
4. Simplify the left side using the property \((b^{m})^{n} = b^{m \cdot n}\), so \(y^{(2\/3) \cdot (3\/2)} = y^{1} = y\).
5. Write the final expression for \(y\): \(y = \left(a^{2\/3} - x^{2\/3}\right)^{3\/2}\).

Key Concepts

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For example, x^(2/3) means the cube root of x squared, or (x^2)^(1/3). Understanding how to manipulate and invert fractional exponents is essential for isolating variables in equations involving them.

Isolating Variables in Equations

Isolating a variable involves rearranging the equation to express that variable explicitly on one side. This often requires inverse operations such as taking roots or powers, especially when variables are raised to fractional exponents, to solve for the specified variable.

Domain Restrictions and Nonzero Denominators

When solving equations with fractional exponents, it is important to consider domain restrictions to avoid undefined expressions, such as division by zero or even roots of negative numbers. The problem states denominators are nonzero, ensuring valid operations during manipulation.