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

Verified step by step guidance

1. Start with the given equation: \(5x^{\frac{3}{4}} - 15 = 0\).
2. Isolate the term with the variable by adding 15 to both sides: \(5x^{\frac{3}{4}} = 15\).
3. Divide both sides by 5 to solve for \(x^{\frac{3}{4}}\): \(x^{\frac{3}{4}} = \frac{15}{5}\).
4. Simplify the right side: \(x^{\frac{3}{4}} = 3\).
5. To solve for \(x\), raise both sides of the equation to the reciprocal power of \(\frac{3}{4}\), which is \(\frac{4}{3}\): \(\left(x^{\frac{3}{4}}\right)^{\frac{4}{3}} = 3^{\frac{4}{3}}\), simplifying to \(x = 3^{\frac{4}{3}}\).

Key Concepts

Solving Equations Involving Rational Exponents

Rational exponents represent roots and powers, such as x^(3/4) meaning the fourth root of x cubed. To solve equations with rational exponents, isolate the term with the exponent and then raise both sides to the reciprocal power to eliminate the exponent.

Isolating the Variable Term

Before applying exponent rules, rearrange the equation to isolate the term containing the variable. This often involves adding, subtracting, multiplying, or dividing both sides of the equation to simplify and prepare for further operations.

Checking for Extraneous Solutions

When solving equations with rational exponents, especially involving even roots, some solutions may not satisfy the original equation. Always substitute solutions back into the original equation to verify their validity.