Find each root. See Example 3. -∜16

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Recognize that the symbol ∜16 represents the fourth root of 16, which means we are looking for a number \( x \) such that \( x^4 = 16 \).

Rewrite 16 as a power of 2: \( 16 = 2^4 \). This helps because the fourth root and the exponent 4 will interact nicely.

Express the fourth root using fractional exponents: \( \sqrt[4]{16} = 16^{\frac{1}{4}} \). Substitute \( 16 = 2^4 \) to get \( (2^4)^{\frac{1}{4}} \).

Use the power of a power rule \( (a^m)^n = a^{m \times n} \) to simplify \( (2^4)^{\frac{1}{4}} = 2^{4 \times \frac{1}{4}} = 2^1 = 2 \).

Since we are solving \( x^4 = 16 \), remember that there are multiple roots: the principal (positive) root and the negative root, as well as complex roots if considering all roots. For real roots, the solutions are \( x = 2 \) and \( x = -2 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Understanding Roots and Radicals

Roots are the inverse operations of powers, where the nth root of a number is a value that, when raised to the nth power, gives the original number. For example, the fourth root (∜) of 16 is a number which raised to the power 4 equals 16.

Evaluating Even-Order Roots of Positive Numbers

When finding even-order roots (like square or fourth roots) of positive numbers, the principal root is always non-negative. For instance, ∜16 equals 2 because 2⁴ = 16, and the principal root is taken as positive.

Handling Negative Signs Outside the Root

A negative sign placed outside the root indicates the entire root value is negated after evaluation. For example, -∜16 means first find ∜16 = 2, then apply the negative sign to get -2.

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