Write each root using exponents and evaluate. ∜-81

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Recognize that the symbol ∜ represents the fourth root, so the expression ∜-81 can be rewritten using exponents as $(-81)^{\frac{1}{4}}$.

Express -81 in terms of its prime factors or powers: since 81 = \$3^4$, rewrite -81 as $-(3^4)$.

Substitute this back into the expression to get $\left(-(3^4)\right)^{\frac{1}{4}}$.

Use the property of exponents that $(a^m)^n = a^{m \times n}$ to simplify the expression inside the parentheses: $\left(-(3^4)\right)^{\frac{1}{4}} = -\left(3^{4 \times \frac{1}{4}}\right)$, noting the negative sign is outside the power.

Simplify the exponent multiplication $4 \times \frac{1}{4} = 1$, so the expression becomes $-3^1$, which simplifies to $-3$.

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주요 개념

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Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, and fourth roots. The symbol ∜ denotes the fourth root, which means finding a number that, when raised to the power of 4, equals the given value. Understanding how to interpret and manipulate these roots is essential for solving the problem.

Expressing Roots Using Exponents

Roots can be rewritten using fractional exponents, where the nth root of a number is expressed as that number raised to the power of 1/n. For example, the fourth root of -81 can be written as (-81)^(1/4). This conversion allows the use of exponent rules to simplify and evaluate the expression.

Evaluating Roots of Negative Numbers

Evaluating roots of negative numbers depends on whether the root is even or odd. Even roots of negative numbers are not real because no real number raised to an even power results in a negative number. Therefore, the fourth root of -81 is not a real number, and understanding this helps determine the nature of the solution.