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: \( \frac{x^{2/3}}{1} + \frac{y^{2/3}}{1} = \frac{a^{2/3}}{1} \).
2. Subtract \( \frac{x^{2/3}}{1} \) from both sides to isolate the term with \( y \): \( \frac{y^{2/3}}{1} = \frac{a^{2/3}}{1} - \frac{x^{2/3}}{1} \).
3. To solve for \( y \), raise both sides of the equation to the power of \( \frac{3}{2} \) to eliminate the fractional exponent: \( y = \left( \frac{a^{2/3} - x^{2/3}}{1} \right)^{3/2} \).
4. Simplify the expression if possible, ensuring that all terms are correctly combined under the power.
5. Verify the solution by substituting back into the original equation to ensure it holds true.

Key Concepts

Solving for a Variable

Solving for a variable involves isolating that variable on one side of the equation. This process often requires algebraic manipulation, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value. In this case, we need to rearrange the equation to express y in terms of x and a.

Exponents and Fractional Powers

Exponents represent repeated multiplication, and fractional powers indicate roots. For example, x^(2/3) means the cube root of x squared. Understanding how to manipulate these powers is crucial for solving equations involving them, as it allows us to rewrite terms in a more manageable form.

Implicit Functions

An implicit function is defined by an equation that relates multiple variables without explicitly solving for one variable. In this case, the equation x^(2/3) + y^(2/3) = a^(2/3) defines y implicitly. Recognizing how to extract y from such equations is essential for finding its explicit form.