In Exercises 29–44, simplify using the quotient rule. _______ ⁵√64x¹⁴/y¹⁵

Verified step by step guidance

1. Identify the expression under the radical: $\frac{64x^{14}}{y^{15}}$.
2. Recognize that the expression is under a fifth root, which can be expressed as ${\left(\frac{64x^{14}}{y^{15}}\right)}^{1/5}$.
3. Apply the quotient rule for exponents: ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$.
4. Simplify the expression by applying the fifth root to both the numerator and the denominator: $\frac{(64x^{14}{)}^{1/5}}{(y^{15}{)}^{1/5}}$.
5. Simplify each part separately: ${64}^{1/5}$, $x^{14/5}$, and $y^{15/5}$.

Key Concepts

Quotient Rule

The quotient rule is a fundamental principle in algebra used to simplify expressions involving division. It states that when dividing two powers with the same base, you subtract the exponents. This rule is essential for simplifying expressions like the one given, where you need to manage the powers of variables in the numerator and denominator.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this question, the fifth root is applied to the expression, which requires understanding how to manipulate and simplify roots. Recognizing how to express numbers and variables in radical form is crucial for simplifying the given expression correctly.

Exponent Rules

Exponent rules govern how to handle powers and roots in algebra. Key rules include the product of powers, power of a power, and the power of a product. These rules are vital for simplifying expressions involving exponents, such as converting between radical and exponential forms, which is necessary for solving the problem at hand.