91Ó°ÊÓ

In mathematics, a cube root of a number is a value that, when multiplied by itself three times, gives the original number. You might recognize this from the symbol \( \sqrt[3]{a} \), which represents the cube root of \(a\).

For example, the cube root of -216 can be found by identifying a number that, when raised to the power of three, equals -216. In this case, -6 is that number because:

\[ (-6) \times (-6) \times (-6) = -216 \].

It's important to note that while square roots of negative numbers result in imaginary numbers, cube roots of negative numbers result in negative real numbers. This is because multiplying three negative numbers still yields a negative result.

Thus, for the exercise: \( (-216)^{\frac{1}{3}} = -6 \)

Negative exponents may seem tricky, but they follow a straightforward rule: \(a^{-b} = \frac{1}{a^b} \). This rule essentially tells us to take the reciprocal of the base raised to the positive exponent.

For example, \( 216^{-\frac{1}{3}} \) can be simplified using this rule. First, rewrite it as the reciprocal:

\[ 216^{-\frac{1}{3}} = \frac{1}{216^{\frac{1}{3}}} \].

We know that \( 216^{\frac{1}{3}} = 6 \), as previously explained. Therefore,

\[ 216^{-\frac{1}{3}} = \frac{1}{6} \].

When working with negative exponents, always remember the key step: transform the expression into a fraction to make the computations easier.

Exponents are a fundamental part of algebra, and understanding their rules helps simplify many mathematical expressions. Here are some essential exponent rules:

• **Product Rule**: \(a^m \times a^n = a^{m+n} \)
• **Quotient Rule**: \(\frac{a^m}{a^n} = a^{m-n} \)
• **Power of a Power**: \((a^m)^n = a^{mn} \)
• **Power of a Product**: \((ab)^n = a^n b^n \)
• **Power of a Quotient**: \((\frac{a}{b})^n = \frac{a^n}{b^n} \)
• **Zero Exponent**: \(a^0 = 1 \), for any nonzero a
• **Negative Exponent**: \(a^{-m} = \frac{1}{a^m} \)

Using these rules can help make sense of more complex problems. For instance, in simplifying \-216^{\frac{1}{3}} \, we applied the rule of handling negative coefficients after evaluating the exponent separately. This ensures accuracy and simplicity in calculations.

Remember, practicing these rules will enhance your confidence and proficiency in working with exponents.