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/(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 \((a^m)^n = a^{m \cdot n}\), to both the numerator and the denominator: numerator becomes \(x^{(2/3) \cdot 2} = x^{4/3}\), denominator becomes \(x^{2 \cdot (7/3)} = x^{14/3}\).
3. Rewrite the expression as a single power of \(x\) by subtracting the exponents in the denominator from the numerator: \(x^{4/3 - 14/3}\).
4. Perform the subtraction in the exponent: \$4/3 - 14/3 = (4 - 14)/3 = -10/3$.
5. Since the problem asks for no negative exponents, rewrite \(x^{-10/3}\) as \(\frac{1}{x^{10/3}}\) to express the final simplified form without negative exponents.

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, dividing exponents when dividing like bases, and adding exponents when multiplying like bases. These rules help simplify complex expressions systematically.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously; for example, x^(m/n) means the nth root of x raised to the mth power. Understanding how to manipulate fractional exponents is essential for simplifying expressions and converting between radical and exponential forms.

Eliminating Negative Exponents

Negative exponents indicate reciprocals, such as x^(-a) = 1/x^a. To write answers without negative exponents, rewrite terms with negative powers as fractions with positive exponents in the denominator. This ensures the expression is in a standard simplified form.