Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (x^1/4y^2/5)²⁰/x²

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Start by applying the power of a power rule to the numerator: \( (x^{\frac{1}{4}} y^{\frac{2}{5}})^{20} \). This means you multiply each exponent inside the parentheses by 20.

Calculate the new exponents for each variable in the numerator: \( x^{\frac{1}{4} \times 20} \) and \( y^{\frac{2}{5} \times 20} \).

Rewrite the expression with the simplified exponents in the numerator: \( x^{5} y^{8} \) (after multiplying the exponents).

Now, write the entire expression as \( \frac{x^{5} y^{8}}{x^{2}} \).

Use the quotient rule for exponents to simplify the \( x \) terms: subtract the exponent in the denominator from the exponent in the numerator, resulting in \( x^{5 - 2} y^{8} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponentiation of Powers

When raising a power to another power, multiply the exponents. For example, (x^(a))^b = x^(a*b). This rule helps simplify expressions like (x^(1/4))^20 by multiplying 1/4 and 20 to get the new exponent.

Division of Exponents with the Same Base

When dividing expressions with the same base, subtract the exponents: x^m / x^n = x^(m-n). This is essential for simplifying the given expression after applying the power rule.

Negative Exponents and Positive Exponent Form

Negative exponents indicate reciprocals, e.g., x^(-a) = 1/x^a. The problem requires answers without negative exponents, so rewrite any negative exponents as positive by moving factors between numerator and denominator.

Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (x1/4y2/5)20/x2