In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. ___ √x⁶y⁷

Verified step by step guidance

1. Identify the expression under the square root: $\sqrt{x^6{y}^7}$.
2. Factor the expression inside the square root into perfect squares: $x^6=(x^3{)}^2$ and $y^7=(y^3{)}^2\cdot y$.
3. Rewrite the square root using the factored form: $\sqrt{(x^3{)}^2(y^3{)}^{2}y}$.
4. Apply the property of square roots: $\sqrt{a^{2}b}=a\sqrt{b}$, to simplify: $x^3{y}^3\sqrt{y}$.
5. The simplified form of the expression is $x^3{y}^3\sqrt{y}$.

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In algebra, this often involves identifying common factors or applying special product formulas, such as the difference of squares or perfect square trinomials. Understanding how to factor is essential for simplifying expressions, especially those involving polynomials or radicals.

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, etc. The radical symbol (√) indicates the root of a number, and simplifying radicals often involves expressing them in their simplest form. This includes removing perfect squares from under the radical sign and rewriting the expression to make calculations easier, which is crucial for solving problems involving roots.

Exponents and Powers

Exponents represent repeated multiplication of a number by itself and are fundamental in algebra for expressing large numbers compactly. Understanding the laws of exponents, such as the product of powers and power of a power, is vital when simplifying expressions with variables raised to powers. In the context of the given expression, recognizing how to manipulate exponents will aid in simplifying the radical expression effectively.