Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (x^2/3)^2/(x^2)^7/3

Verified step by step guidance

1. Start by simplifying the expression $\frac{(x^{2/3}{)}^2}{(x^2{)}^{7/3}}$.
2. Apply the power of a power property $(a^m{)}^n=a^{m\cdot n}$ to both the numerator and the denominator.
3. For the numerator: $(x^{2/3}{)}^2=x^{(2/3)\cdot 2}=x^{4/3}$.
4. For the denominator: $(x^2{)}^{7/3}=x^{2\cdot (7/3)}=x^{14/3}$.
5. Now, simplify the expression by using the property $\frac{a^m}{a^n}=a^{m-n}$, resulting in $x^{4/3-14/3}$.

Key Concepts

Exponent Rules

Exponent rules are fundamental principles that govern how to manipulate expressions involving powers. Key rules 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 rules is essential for simplifying expressions with exponents.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/(a^n). In the context of simplification, it is important to rewrite any negative exponents as positive ones to adhere to the problem's requirement of expressing answers without negative exponents.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. This process often requires applying exponent rules and ensuring that all variables are treated as positive real numbers, which helps avoid complications with undefined expressions.