Solve the Rational Equation for x fourth root of 8x+12-9=-11
The problem is a mathematical equation that involves rational expressions and an operation with a radical. Specifically, the equation features the fourth root of an expression, 8x + 12, which is then subtracted by 9. The result of this operation is given as equal to -11. The task is to find the value(s) of the variable x that satisfy this equation. To solve this kind of equation, one typically isolates the radical expression on one side of the equation and then raises both sides of the equation to the power that cancels out the radical, in this case, the fourth power. This should be followed by simplifying the resulting equation and finding the value of x.
$\sqrt[4]{8 x + 12} - 9 = - 11$
Answer
Solution:
Step 1: Isolate the fourth root term
Move all terms not involving $\sqrt[4]{8x + 12}$ to the opposite side.
- Add $9$ to both sides: $\sqrt[4]{8x + 12} = -11 + 9$
- Combine like terms: $\sqrt[4]{8x + 12} = -2$
Step 2: Eliminate the fourth root
Raise both sides to the fourth power to remove the fourth root.
- $(\sqrt[4]{8x + 12})^4 = (-2)^4$
Step 3: Simplify both sides
- Rewrite the fourth root as a power: $(8x + 12)^{\frac{1}{4}}$
- Apply the power of a power rule: $(8x + 12)^{\frac{1}{4} \cdot 4} = (-2)^4$
- Cancel out the powers: $(8x + 12)^1 = (-2)^4$
- Simplify the right-hand side: $8x + 12 = 16$
Step 4: Solve for $x$
- Subtract $12$ from both sides: $8x = 16 - 12$
- Simplify: $8x = 4$
- Divide by $8$: $x = \frac{4}{8}$
Step 5: Simplify the solution
- Reduce the fraction: $x = \frac{1}{2}$
Step 6: Check the solution
- Verify that the solution does not make the original equation undefined: No solution is excluded.
Knowledge Notes:
Rational Equations: Equations that involve rational expressions, which are ratios of polynomials.
Radicals: Expressions that contain roots, such as square roots or fourth roots.
Isolating the Radical: The process of moving all terms without the radical to the other side of the equation.
Exponent Rules: Important rules include the power of a power rule $(a^m)^n = a^{mn}$ and the power of a product rule $(ab)^n = a^n b^n$.
Simplifying Expressions: Combining like terms, reducing fractions, and canceling common factors.
Checking Solutions: Substituting the solution back into the original equation to ensure it does not result in an undefined expression or a contradiction.
