Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. (x^2 - x - 4)^(3/4) - 2 = 6

Verified step by step guidance

1. Start by isolating the expression with the rational exponent on one side of the equation. Add 2 to both sides to get: $(x^2-x-4{)}^{3/4}=8$.
2. To eliminate the rational exponent, raise both sides of the equation to the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$. This gives: $(x^2-x-4)=8^{4/3}$.
3. Calculate $8^{4/3}$ by first finding the cube root of 8, which is 2, and then raising it to the power of 4, resulting in $2^4$.
4. Set the equation $x^2-x-4=16$ and solve for $x$ by rearranging it into a standard quadratic form: $x^2-x-20=0$.
5. Use the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ where $a=1$, $b=-1$, and $c=-20$ to find the values of $x$.

Key Concepts

Rational Exponents

Rational exponents are a way to express roots and powers using fractions. For example, an exponent of 1/n indicates the n-th root of a number, while m/n represents the n-th root raised to the m-th power. Understanding how to manipulate these exponents is crucial for solving equations involving them, as they can be rewritten in radical form.

Solving Polynomial Equations

Solving polynomial equations involves finding the values of the variable that make the equation true. This often requires rearranging the equation, factoring, or using the quadratic formula. In the context of the given equation, isolating the polynomial expression and then applying appropriate algebraic techniques is essential for finding the solutions.

Checking Solutions

Checking solutions is a critical step in verifying that the proposed answers satisfy the original equation. This involves substituting the found values back into the equation to ensure both sides are equal. This step is particularly important when dealing with rational exponents, as extraneous solutions may arise during the solving process.