Write each root using exponents and evaluate. ∜81

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Recognize that the symbol ∜ represents the fourth root, so ∜81 means the fourth root of 81.

Rewrite the fourth root using exponents: the fourth root of a number \(a\) can be written as \(a^{\frac{1}{4}}\). So, \(\sqrt[4]{81} = 81^{\frac{1}{4}}\).

Express 81 as a power of a smaller base if possible. Since \(81 = 3^4\), rewrite the expression as \( (3^4)^{\frac{1}{4}} \).

Use the power of a power property \( (a^m)^n = a^{m \times n} \) to simplify: \( (3^4)^{\frac{1}{4}} = 3^{4 \times \frac{1}{4}} \).

Multiply the exponents: \(4 \times \frac{1}{4} = 1\), so the expression simplifies to \(3^1\), which is just 3.

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주요 개념

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Radicals and Roots

Radicals represent roots of numbers, 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 81 is the number that raised to the 4th power equals 81.

Expressing Roots Using Exponents

Roots can be rewritten as fractional exponents: the nth root of a number is the same as raising that number to the power of 1/n. For instance, ∜81 can be expressed as 81^(1/4), which simplifies calculations using exponent rules.

Evaluating Powers and Roots

To evaluate expressions like 81^(1/4), factor the base into prime factors and apply the fractional exponent to each. Since 81 = 3^4, raising it to the 1/4 power simplifies to 3^(4*(1/4)) = 3, giving the value of the root.