Solve for x: $\(\sqrt\)[3]{x \(\sqrt{x}\)} = 9$

Verified step by step guidance

1. Rewrite the expression inside the cube root using exponents. Recall that \( \sqrt{x} = x^{\frac{1}{2}} \), so \( x \sqrt{x} = x \cdot x^{\frac{1}{2}} = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}} \).
2. Express the cube root as an exponent: \( \sqrt[3]{x^{\frac{3}{2}}} = \left(x^{\frac{3}{2}}\right)^{\frac{1}{3}} \).
3. Use the power of a power property to simplify the exponent: \( \left(x^{\frac{3}{2}}\right)^{\frac{1}{3}} = x^{\frac{3}{2} \cdot \frac{1}{3}} = x^{\frac{1}{2}} \).
4. Set the simplified expression equal to 9: \( x^{\frac{1}{2}} = 9 \).
5. To solve for \( x \), square both sides to eliminate the fractional exponent: \( \left(x^{\frac{1}{2}}\right)^2 = 9^2 \), which simplifies to \( x = 81 \).

Key Concepts

Radical Expressions and Their Properties

Radical expressions involve roots such as square roots and cube roots. Understanding how to manipulate and simplify these expressions, including converting between radicals and exponents, is essential for solving equations involving roots.

Exponents and Fractional Powers

Exponents can be fractions, where the numerator represents the power and the denominator the root. For example, x^(m/n) means the nth root of x raised to the mth power. This concept helps rewrite radicals as exponents to simplify and solve equations.

Solving Equations Involving Radicals

To solve equations with radicals, isolate the radical expression and then eliminate the root by raising both sides to the appropriate power. This process often requires careful handling of exponents and checking for extraneous solutions.