In Exercises 45–66, divide and, if possible, simplify. ______ √54a⁷b¹¹ √3a⁻⁴b⁻²

Verified step by step guidance

1. Step 1: Start by writing the expression as a single fraction under a square root: $\frac{\sqrt{54a^7{b}^{11}}}{\sqrt{3a^{-4}{b}^{-2}}}$.
2. Step 2: Use the property of square roots that $\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}$ to combine the expression under one square root: $\sqrt{\frac{54a^7{b}^{11}}{3a^{-4}{b}^{-2}}}$.
3. Step 3: Simplify the fraction inside the square root by dividing the coefficients and using the laws of exponents: $\frac{54}{3}=18$, $a^{7-(-4)}=a^{7+4}=a^{11}$, and $b^{11-(-2)}=b^{11+2}=b^{13}$.
4. Step 4: Rewrite the expression inside the square root: $\sqrt{18a^{11}{b}^{13}}$.
5. Step 5: Simplify the square root by factoring out perfect squares: $\sqrt{18}=\sqrt{9\times 2}=3\sqrt{2}$, $\sqrt{a^{11}}=a^5\sqrt{a}$, and $\sqrt{b^{13}}=b^6\sqrt{b}$. Combine these to express the simplified form.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, the square root of a number or variable is being simplified. Understanding how to manipulate these expressions is crucial for performing operations like division and simplification.

Properties of Exponents

The properties of exponents govern how to handle expressions involving powers. Key rules include the product of powers, quotient of powers, and power of a power. These rules are essential for simplifying expressions with variables raised to exponents, especially when dividing terms.

Simplification of Fractions

Simplifying fractions involves reducing them to their lowest terms by canceling common factors. In the context of radical expressions, this means identifying and removing any common factors in the numerator and denominator, which is necessary for achieving a simplified final answer.