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

Verified step by step guidance

1. Start by isolating the term with the variable. Add 15 to both sides of the equation: $5x^{3/4}=15$.
2. Divide both sides by 5 to solve for $x^{3/4}$: $x^{3/4}=3$.
3. To eliminate the fractional exponent, raise both sides of the equation to the power of $4/3$: $(x^{3/4}{)}^{4/3}=3^{4/3}$.
4. Simplify the left side of the equation using the property $(a^m{)}^n=a^{m\cdot n}$, which results in $x=3^{4/3}$.
5. The solution is $x=3^{4/3}$, which can be further simplified if needed.

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.