Find each root. ∜(5 + 2m)⁴

Verified step by step guidance

1. Recognize that the expression is a fourth root (∜) of the quantity (5 + 2m) raised to the fourth power, which can be written as \(\sqrt[4]{(5 + 2m)^4}\).
2. Recall the property of radicals and exponents: \(\sqrt[n]{a^n} = |a|\) when n is even, because the even root of an even power results in the absolute value of the base.
3. Apply this property to simplify \(\sqrt[4]{(5 + 2m)^4}\) to \(|5 + 2m|\).
4. Understand that the root of the expression is the absolute value of the inner expression, so the roots are the values of \$5 + 2m$ that satisfy the equation when set equal to zero.
5. Set \$5 + 2m = 0\( and solve for \)m\( by isolating \)m\(: subtract 5 from both sides and then divide by 2, resulting in \)m = -\frac{5}{2}$.

Key Concepts

Understanding Roots and Exponents

Roots and exponents are inverse operations. The fourth root (∜) of a number is the value that, when raised to the fourth power, gives the original number. Recognizing how to simplify expressions involving roots and powers is essential for solving the problem.

Properties of Exponents

When an expression with an exponent is raised to a root, the exponent and root interact multiplicatively. Specifically, the nth root of a quantity raised to the mth power equals the quantity raised to the m/n power. This property helps simplify ∜(5 + 2m)⁴ to (5 + 2m)^(4/4).

Simplifying Algebraic Expressions

Simplifying expressions involves reducing them to their simplest form by applying algebraic rules. In this problem, after applying the root and exponent properties, simplifying the resulting expression correctly is necessary to find the root.