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

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Start by rewriting the expression clearly: \(\frac{(x^{2/3})^{2}}{(x^{2})^{7/3}}\).

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 \(x^{2 \times (7/3)}\).

Simplify the exponents by multiplying: numerator exponent is \(\frac{2}{3} \times 2 = \frac{4}{3}\), denominator exponent is \(2 \times \frac{7}{3} = \frac{14}{3}\).

Rewrite the expression as a single power of \(x\) by subtracting the exponent in the denominator from the exponent in the numerator: \(x^{\frac{4}{3} - \frac{14}{3}}\).

Simplify the exponent subtraction and rewrite the expression without negative exponents by using the property \(x^{-m} = \frac{1}{x^{m}}\) if needed.

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주요 개념

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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).

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.

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

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주요 개념

Laws of Exponents

Rational Exponents

Eliminating Negative Exponents

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