Solve each equation. 3x^3/4 = x^1/2

Verified step by step guidance

1. Rewrite the equation \$3x^{3/4} = x^{1/2}\( and recognize that both sides have expressions with exponents involving \)x$.
2. To solve for \(x\), first bring all terms to one side to set the equation equal to zero: \$3x^{3/4} - x^{1/2} = 0$.
3. Factor out the common term with the smallest exponent, which is \(x^{1/2}\), so the equation becomes \(x^{1/2}(3x^{1/4} - 1) = 0\).
4. Set each factor equal to zero separately: \(x^{1/2} = 0\) and \$3x^{1/4} - 1 = 0$.
5. Solve each equation for \(x\): For \(x^{1/2} = 0\), raise both sides to the power of 2; for \$3x^{1/4} - 1 = 0\(, isolate \)x^{1/4}$ and then raise both sides to the power of 4.

Key Concepts

Exponents and Rational Exponents

Exponents indicate repeated multiplication, and rational exponents represent roots; for example, x^(1/2) means the square root of x. Understanding how to manipulate and simplify expressions with fractional exponents is essential for solving equations involving roots and powers.

Solving Equations Involving Exponents

To solve equations with exponents, isolate the variable term and use properties of exponents to rewrite expressions with a common base or exponent. This often involves raising both sides to a power that eliminates the fractional exponent or rewriting terms to compare powers directly.

Domain Restrictions for Variables with Rational Exponents

When variables have fractional exponents with even denominators, the base must be non-negative to keep the expression real. Recognizing domain restrictions ensures solutions are valid within the real number system and prevents extraneous or undefined solutions.