Solve: 5x^3/4- 15 = 0.

Verified step by step guidance

1. Isolate the term containing the variable by adding 15 to both sides of the equation: \(5x^{3/4} - 15 + 15 = 0 + 15\), which simplifies to \(5x^{3/4} = 15\).
2. Divide both sides of the equation by 5 to solve for \(x^{3/4}\): \(\frac{5x^{3/4}}{5} = \frac{15}{5}\), which simplifies to \(x^{3/4} = 3\).
3. To eliminate the fractional exponent \(3/4\), raise both sides of the equation to the reciprocal power \(4/3\): \((x^{3/4})^{4/3} = 3^{4/3}\). Using the property \((a^{m})^{n} = a^{m \cdot n}\), the left-hand side simplifies to \(x\), so \(x = 3^{4/3}\).
4. Simplify \(3^{4/3}\) if needed. This can be expressed as \((3^{1/3})^4\), where \(3^{1/3}\) represents the cube root of 3, raised to the fourth power.
5. State the solution as \(x = 3^{4/3}\), or approximate the value if required, depending on the context of the problem.

Key Concepts

Exponents and Fractional Powers

Exponents represent repeated multiplication, and fractional exponents indicate roots. For example, x^(3/4) means the fourth root of x cubed. Understanding how to manipulate and simplify expressions with fractional exponents is crucial for solving equations involving them.

Isolating Variables

Isolating a variable involves rearranging an equation to solve for that variable. This often includes adding, subtracting, multiplying, or dividing both sides of the equation by the same value. In the given equation, isolating x requires moving constants to one side and simplifying the expression.

Roots and Their Properties

Finding roots of an equation involves determining the values of the variable that satisfy the equation. In this case, solving for x requires understanding how to apply the properties of roots, including how to handle both positive and negative roots, especially when dealing with even and odd roots.