Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (x^2/3)²/(x²)^7/3

Verified step by step guidance

1. Start by rewriting the expression clearly: \(\frac{(x^{2/3})^{2}}{(x^{2})^{7/3}}\).
2. Apply the power of a power rule, which states \((a^{m})^{n} = a^{m \times n}\), to both the numerator and the denominator: numerator becomes \(x^{(2/3) \times 2}\) and denominator becomes \(x^{2 \times (7/3)}\).
3. Simplify the exponents by multiplying: numerator exponent is \(\frac{2}{3} \times 2 = \frac{4}{3}\), denominator exponent is \(2 \times \frac{7}{3} = \frac{14}{3}\).
4. Rewrite the expression as a single power of \(x\) by subtracting the exponent in the denominator from the exponent in the numerator: \(x^{\frac{4}{3} - \frac{14}{3}}\).
5. Simplify the exponent subtraction and rewrite the expression without negative exponents by using the property \(x^{-m} = \frac{1}{x^{m}}\) if needed.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include multiplying exponents when raising a power to another power, and subtracting exponents when dividing like bases. For example, (x^a)^b = x^(a*b) and x^m / x^n = x^(m-n).

Rational Exponents

Rational exponents represent roots and powers simultaneously, where x^(m/n) means the nth root of x raised to the mth power. Understanding how to manipulate these exponents is essential for simplifying expressions with fractional powers.

Eliminating Negative Exponents

Negative exponents indicate reciprocals, such that x^(-a) = 1/x^a. To write answers without negative exponents, rewrite terms with negative powers as fractions with positive exponents in the denominator.