Problem 12 - Intermediate Algebra

Question

For Problems \(1-30\), evaluate each numerical expression. $$ \left(-\frac{8}{27}\right)^{-\frac{1}{3}} $$

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{3}{2}\).

Step 1: Understanding the Problem

We need to simplify the expression \( \left(-\frac{8}{27}\right)^{-\frac{1}{3}} \). This involves applying the rules of exponents and roots.

Step 2: Handling the Negative Exponent

The negative exponent indicates that we must find the reciprocal of the base raised to the absolute value of the exponent. Thus, \( \left(-\frac{8}{27}\right)^{-\frac{1}{3}} \) becomes \( \frac{1}{\left(-\frac{8}{27}\right)^{\frac{1}{3}}} \).

Step 3: Evaluating the Rational Exponent

A rational exponent like \( \frac{1}{3} \) means a cube root. Therefore, we need to find the cube root of \( -\frac{8}{27} \).

Step 4: Finding the Cube Root

Cube root the numerical values separately: \( \sqrt[3]{-8} = -2 \) and \( \sqrt[3]{27} = 3 \).Thus, \( \sqrt[3]{-\frac{8}{27}} = -\frac{2}{3} \).

Step 5: Final Calculation

Putting it all together, we now substitute back: \( \frac{1}{-\frac{2}{3}} = -\frac{3}{2} \).

Step 6: Conclusion

The value of the expression \( \left(-\frac{8}{27}\right)^{-\frac{1}{3}} \) is -\( \frac{3}{2} \).

Key Concepts

Rational Exponents

Rational exponents are expressions where the exponent is a fraction, like \( a^{m/n} \). When dealing with rational exponents, you can think of them as a combination of powers and roots. The denominator of the fraction (\( n \)) corresponds to the root, while the numerator (\( m \)) represents the power. For example, if you have \( x^{1/3} \), it means taking the cube root of \( x \). Similarly, \( x^{2/3} \) means you first take the cube root of \( x \) and then square the result.
Here are some quick pointers to simplify expressions with rational exponents:

• If \( x^{1/n} \), calculate the \( n \)-th root of \( x \).
• For \( x^{m/n} \), first find \( x^{1/n} \) and then raise it to the \( m \)-th power.

Using rational exponents allows more flexibility and can simplify expressions with roots and powers.

Negative Exponents

Negative exponents can often be tricky but are simple once you understand the concept. A negative exponent, like \( x^{-a} \), means the reciprocal of the base raised to the absolute value of the exponent. In simpler terms, \( x^{-a} \) transforms to \( \frac{1}{x^a} \). For example, if you have \( 3^{-2} \), it is equal to \( \frac{1}{3^2} = \frac{1}{9} \).
Here is a step-by-step approach to handling negative exponents:

• Identify the base and the negative exponent.
• Calculate the reciprocal of the base.
• Raise the reciprocal to the absolute value of the negative exponent.

Properly managing negative exponents is essential for simplifying complex expressions and solving them efficiently.

Cube Root

The cube root is a special type of root signified by \( \sqrt[3]{x} \). It represents a value that, when multiplied by itself three times, gives the original number \( x \). For instance, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \). Similarly, the cube root of -8 is -2 since \( -2 \times -2 \times -2 = -8 \).
To find a cube root, you follow these steps:

• Identify the number for which you need to find the cube root.
• Split the number into its logical parts or smaller numbers, if necessary, that can be easily cubed back.
• Calculate using any necessary multipliers to verify the cube returns to the original number.

Understanding the application of cube roots helps in solving many algebraic expressions and equations.