Find each root. ∜81p¹²q⁴

Verified step by step guidance

1. Identify the expression under the fourth root: \(81p^{12}q^4\).
2. Recognize that taking the fourth root is equivalent to raising each factor inside the radical to the power of \(\frac{1}{4}\).
3. Break down the expression into separate components: \(81\), \(p^{12}\), and \(q^4\).
4. Calculate the fourth root of each component: \(\sqrt[4]{81}\), \(\sqrt[4]{p^{12}}\), and \(\sqrt[4]{q^4}\).
5. Combine the results to express the fourth root of the entire expression.

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, and higher-order roots. The notation ∜81p¹²q⁴ indicates the fourth root of the expression 81p¹²q⁴. Understanding how to simplify radicals is essential for finding roots, as it involves breaking down the expression into its prime factors and applying the properties of exponents.

Exponents and Powers

Exponents represent repeated multiplication of a base number. In the expression 81p¹²q⁴, the exponent indicates how many times the base is multiplied by itself. Knowing the rules of exponents, such as how to handle powers when taking roots, is crucial for simplifying expressions and finding roots accurately.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms and applying the properties of exponents and radicals. For the expression ∜81p¹²q⁴, this means calculating the fourth root of each component separately and expressing the result in a clear and concise manner. Mastery of simplification techniques is vital for solving algebraic problems efficiently.