In Exercises 91–100, simplify using properties of exponents. (x^2/3)^3

Verified step by step guidance

1. Identify the expression: $(x^{2/3}{)}^3$.
2. Apply the power of a power property of exponents, which states $(a^m{)}^n=a^{m\cdot n}$.
3. Multiply the exponents: $\frac{2}{3}\cdot 3$.
4. Simplify the multiplication of the exponents.
5. Rewrite the expression with the simplified exponent.

Key Concepts

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ((a^m)^n = a^(m*n)), and the quotient of powers (a^m / a^n = a^(m-n)). Understanding these properties is essential for simplifying expressions with exponents.

Power of a Power Rule

The power of a power rule states that when raising a power to another power, you multiply the exponents. For example, (a^m)^n simplifies to a^(m*n). This rule is crucial for simplifying expressions like (x^(2/3))^3, as it allows you to combine the exponents effectively.

Fractional Exponents

Fractional exponents represent roots in addition to powers. For instance, an exponent of 1/2 corresponds to the square root, while 2/3 indicates the cube root of the square. Understanding how to interpret and manipulate fractional exponents is vital for simplifying expressions that involve them, such as (x^(2/3))^3.