Find each root. See Example 3. -∜16

검증된 단계별 안내

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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주요 개념

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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.