Solve each equation. See Example 7. 3x^3/4 = x^1/2

Verified step by step guidance

1. Rewrite the equation \$3x^{3/4} = x^{1/2}$ to have the same base and express both sides with exponents clearly.
2. Recall that \(x^{1/2}\) is the same as \(x^{2/4}\), so rewrite the equation as \$3x^{3/4} = x^{2/4}$.
3. Divide both sides of the equation by \(x^{2/4}\) to isolate the terms with \(x\) on one side: \$3x^{3/4 - 2/4} = 1$.
4. Simplify the exponent subtraction: \$3x^{1/4} = 1$.
5. Solve for \(x^{1/4}\) by dividing both sides by 3, then raise both sides of the equation to the 4th power to solve for \(x\).

Key Concepts

Exponents and Rational Exponents

Exponents indicate how many times a base is multiplied by itself. Rational exponents, such as fractions, represent roots; for example, x^(1/2) means the square root of x, and x^(3/4) means the fourth root of x cubed. Understanding how to manipulate and interpret these is essential for solving equations involving fractional powers.

Properties of Exponents

Properties like the product rule (a^m * a^n = a^(m+n)) and power rule ((a^m)^n = a^(m*n)) help simplify expressions with exponents. These rules allow rewriting and comparing terms with exponents, which is crucial when solving equations where variables have different fractional exponents.

Solving Equations Involving Exponents

To solve equations with exponents, isolate the variable term and use inverse operations such as raising both sides to a power that eliminates the fractional exponent. This often involves rewriting the equation to have the same base or exponent, enabling you to solve for the variable systematically.