Simplify using properties of exponents.$(x^{\frac{2}{3}}{)}^3$

Verified step by step guidance

1. Identify the expression to simplify: \(\left( x^{\frac{2}{3}} \right)^3\).
2. Recall the power of a power property of exponents: \((a^m)^n = a^{m \times n}\).
3. Apply this property by multiplying the exponents: \(x^{\frac{2}{3} \times 3}\).
4. Simplify the exponent multiplication: \(\frac{2}{3} \times 3 = 2\).
5. Rewrite the expression with the simplified exponent: \(x^2\).

Key Concepts

Properties of Exponents

Properties of exponents are rules that simplify expressions involving powers, such as the product rule, quotient rule, and power rule. These rules help combine or break down exponents to simplify expressions efficiently.

Power of a Power Rule

The power of a power rule states that when raising an exponent to another exponent, you multiply the exponents. For example, (x^a)^b = x^(a*b), which is essential for simplifying expressions like (x^(2/3))^3.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously, where the numerator is the power and the denominator is the root. For example, x^(2/3) means the cube root of x squared, which is important for understanding and simplifying expressions with fractional exponents.