Match each equation in Column I with the correct first step for solving it in Column II. (x+5)^2/3 - (x+5)^1/3 - 6 = 0

Verified step by step guidance

1. Identify the substitution to simplify the equation. Let \( y = (x+5)^{1/3} \), so that \( y^2 = (x+5)^{2/3} \).
2. Rewrite the original equation in terms of \( y \) using the substitution: \( y^2 - y - 6 = 0 \).
3. Recognize that the rewritten equation is a quadratic in \( y \), which can be solved using factoring, completing the square, or the quadratic formula.
4. Solve the quadratic equation for \( y \) to find the possible values of \( y \).
5. Back-substitute \( y = (x+5)^{1/3} \) and solve for \( x \) by cubing both sides of the equation for each value of \( y \).

Key Concepts

Exponents and Rational Powers

Understanding exponents, especially rational exponents like 2/3 and 1/3, is crucial. These represent roots and powers combined, for example, x^(2/3) means the cube root of x squared. Recognizing how to manipulate and simplify expressions with rational exponents helps in rewriting and solving the equation.

Substitution Method

The substitution method involves replacing a complex expression with a simpler variable to make the equation easier to solve. Here, letting y = (x+5)^(1/3) transforms the equation into a quadratic form in terms of y, simplifying the solving process.

Solving Quadratic Equations

Once the substitution is made, the resulting equation is quadratic. Knowing how to solve quadratic equations using factoring, completing the square, or the quadratic formula is essential to find the values of the substituted variable, which can then be back-substituted to find x.