Write each root using exponents and evaluate. ∛-343

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Recognize that the cube root symbol ∛ can be rewritten using rational exponents. Specifically, the cube root of a number \(a\) is expressed as \(a^{\frac{1}{3}}\). So, rewrite \(\sqrt[3]{-343}\) as \((-343)^{\frac{1}{3}}\).

Identify the base number inside the root, which is \(-343\). Note that \(343\) is a perfect cube because \(7^3 = 343\).

Since the base is negative, recall that the cube root of a negative number is the negative of the cube root of the positive number. This is because cube roots preserve the sign of the original number.

Rewrite \((-343)^{\frac{1}{3}}\) as \(- (343)^{\frac{1}{3}}\) to separate the negative sign from the positive cube root.

Evaluate \((343)^{\frac{1}{3}}\) by finding the number which, when raised to the power of 3, equals 343. Since \(7^3 = 343\), \((343)^{\frac{1}{3}} = 7\). Therefore, the cube root of \(-343\) is \(-7\).

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Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, etc. The cube root (∛) of a number is the value that, when multiplied by itself three times, gives the original number. Understanding how to interpret and manipulate these roots is essential for solving the problem.

Exponents and Fractional Exponents

Roots can be expressed using fractional exponents, where the nth root of a number is written as that number raised to the power of 1/n. For example, the cube root of a number is the same as raising it to the 1/3 power. This allows for easier manipulation and evaluation using exponent rules.

Evaluating Cube Roots of Negative Numbers

Unlike even roots, cube roots of negative numbers are real and negative because a negative number multiplied three times remains negative. For example, ∛-343 equals -7 since (-7)³ = -343. Recognizing this helps correctly evaluate roots of negative values.