Exercises 177–179 will help you prepare for the material covered in the next section. If $-8$ is substituted for x in the equation $5x^{\frac{2}{3}}+11x^{\frac{1}{3}}+2=0$, is the resulting statement true or false?

Verified step by step guidance

1. First, recognize that the equation given is \$5x^{\frac{2}{3}} + 11x^{\frac{1}{3}} + 2 = 0\(. We need to substitute \)x = -8$ into this equation and check if the resulting statement is true or false.
2. Next, calculate \(x^{\frac{1}{3}}\) when \(x = -8\). Since the exponent \(\frac{1}{3}\) represents the cube root, find \(\sqrt[3]{-8}\).
3. Then, use the value of \(x^{\frac{1}{3}}\) to find \(x^{\frac{2}{3}}\). Recall that \(x^{\frac{2}{3}} = \left(x^{\frac{1}{3}}\right)^2\).
4. Substitute the values of \(x^{\frac{2}{3}}\) and \(x^{\frac{1}{3}}\) back into the original equation: \$5x^{\frac{2}{3}} + 11x^{\frac{1}{3}} + 2$.
5. Finally, simplify the expression step-by-step and determine if the result equals zero. If it does, the statement is true; if not, it is false.

Key Concepts

Evaluating Expressions with Rational Exponents

Rational exponents represent roots and powers, where an exponent like 2/3 means taking the cube root first, then squaring the result. Understanding how to correctly evaluate expressions with fractional exponents is essential for substituting values into the equation.

Substitution in Algebraic Equations

Substitution involves replacing the variable with a given value to simplify or evaluate the expression. This process helps determine if the equation holds true for that specific value, which is key to answering whether the statement is true or false.

Checking the Validity of an Equation

After substitution, simplifying both sides of the equation and comparing them determines if the equation is true or false. This concept ensures that the solution or evaluation is accurate and consistent with algebraic principles.