Solve for x cube root of (3x+8)^2=1

Solution:

Step 1:

Cube both sides to eliminate the cube root: $(\sqrt[3]{(3x+8)^2})^3 = 1^3$.

Step 2:

Simplify the equation.

Step 2.1:

Express the cube root as a fractional exponent: $(3x+8)^{\frac{2}{3}}$.

Step 2.2:

Simplify the left side by applying exponent rules.

Step 2.2.1:

Use the power of a power rule: $(a^m)^n = a^{mn}$.

Step 2.2.1.1:

Apply the rule to the expression: $(3x+8)^{\frac{2}{3} \cdot 3} = 1^3$.

Step 2.2.1.2:

Reduce the exponents: $(3x+8)^{\frac{2}{\cancel{3}} \cdot \cancel{3}} = 1^3$.

Step 2.2.1.2.1:

Simplify the expression: $(3x+8)^2 = 1^3$.

Step 2.3:

Simplify the right side: $1$ raised to any power is $1$.

Step 3:

Solve for $x$.

Step 3.1:

Take the square root of both sides: $3x + 8 = \pm\sqrt{1}$.

Step 3.2:

Recognize that the square root of $1$ is $1$.

Step 3.3:

Consider both the positive and negative square roots for the complete solution.

Step 3.3.1:

Solve using the positive root: $3x + 8 = 1$.

Step 3.3.2:

Isolate $x$ by moving other terms to the other side.

Step 3.3.2.1:

Subtract $8$ from both sides: $3x = 1 - 8$.

Step 3.3.2.2:

Combine like terms: $3x = -7$.

Step 3.3.3:

Divide by $3$ to solve for $x$.

Step 3.3.3.1:

Divide both sides by $3$: $\frac{3x}{3} = \frac{-7}{3}$.

Step 3.3.3.2:

Simplify the equation: $x = -\frac{7}{3}$.

Step 3.3.4:

Solve using the negative root: $3x + 8 = -1$.

Step 3.3.5:

Isolate $x$ by moving other terms to the other side.

Step 3.3.5.1:

Subtract $8$ from both sides: $3x = -1 - 8$.

Step 3.3.5.2:

Combine like terms: $3x = -9$.

Step 3.3.6:

Divide by $3$ to solve for $x$.

Step 3.3.6.1:

Divide both sides by $3$: $\frac{3x}{3} = \frac{-9}{3}$.

Step 3.3.6.2:

Simplify the equation: $x = -3$.

Step 3.3.7:

The complete solution includes both values: $x = -\frac{7}{3}, -3$.

Step 4:

Present the result in various forms.

Exact Form: $x = -\frac{7}{3}, -3$

Decimal Form: $x \approx -2.333, -3$

Mixed Number Form: $x = -2\frac{1}{3}, -3$

Knowledge Notes:

The problem involves solving for $x$ in the equation where $x$ is within a cube root and squared. The steps to solve such an equation include:

1. Removing the radical by raising both sides of the equation to the power that corresponds to the root (cubing both sides in this case).

2. Simplifying the equation by applying exponent rules, such as the power of a power rule, which states that $(a^m)^n = a^{mn}$.

3. Taking the square root of both sides to solve for the variable, considering both the positive and negative roots since squaring a number always results in a positive value.

4. Isolating the variable on one side of the equation by performing algebraic operations such as addition, subtraction, multiplication, or division.

5. Presenting the solution in different forms, including exact form, decimal form, and mixed number form.

Understanding these algebraic principles and rules is essential for solving equations involving radicals and exponents.