Problem 12 - Algebra for College Students

Question

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: Recognize the Negative Exponent

The expression given is \( \left(-\frac{8}{27}\right)^{-\frac{1}{3}} \). A negative exponent indicates that we need to take the reciprocal of the base and then apply the positive exponent. Thus, the expression becomes \( \left(-\frac{27}{8}\right)^{\frac{1}{3}} \).

Step 2: Apply the Cube Root

The expression is now \( \left(-\frac{27}{8}\right)^{\frac{1}{3}} \). The exponent \( \frac{1}{3} \) signifies finding the cube root. Therefore, take the cube root of \( -\frac{27}{8} \).

Step 3: Find the Cube Root of the Numerator

Find the cube root of \(-27\). Since \(-3^3 = -27\), the cube root of \(-27\) is \(-3\).

Step 4: Find the Cube Root of the Denominator

Find the cube root of \(8\). Since \(2^3 = 8\), the cube root of \(8\) is \(2\).

Step 5: Combine the Results

Combine the results from Step 3 and Step 4 to express \( \left(-\frac{27}{8}\right)^{\frac{1}{3}} \) as \( \frac{-3}{2} \). This is the simplified answer to the problem.

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they simply involve a change in perspective. Think of a negative exponent as a command to "flip" the base. In mathematical terms, flipping the base means taking its reciprocal. For example, if you have a term like \( a^{-n} \), you actually need to rewrite it as \( \frac{1}{a^n} \).

• The negative sign in the exponent tells you to invert the base (take the reciprocal).
• Once the base is flipped, you then raise it to the positive exponent.

In the original exercise, the expression \( \left(-\frac{8}{27}\right)^{-\frac{1}{3}} \) is transformed by flipping the fraction to become \( \left(-\frac{27}{8}\right)^{\frac{1}{3}} \). This step sets the stage for further simplification using cube roots.

Cube Roots

Cube roots help us find the number that, when multiplied by itself twice, results in the original number. In simpler terms, the cube root of a number \( x \) is a value \( y \) such that \( y^3 = x \). This means we're looking for a number which can be multiplied two more times to reproduce the original value.

• The cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).
• The cube root of -27 is -3 because \( (-3) \times (-3) \times (-3) = -27 \).

Applying cube roots to fractions means you consider the numerator and the denominator separately. For \( \left(-\frac{27}{8}\right)^{\frac{1}{3}} \), evaluate the cube root of the numerator \(-27\) and the denominator \(8\) individually, resulting in \( -3 \) and \( 2 \), respectively. This leads you to \( \frac{-3}{2} \).

Fractional Exponents

Fractional exponents are a unique way to express roots. The expression \( x^{m/n} \) is equivalent to taking the \( n \)-th root of \( x \) and then raising that result to the power of \( m \). This can also be written as \( (\sqrt[n]{x})^m \).

• The numerator tells you what power to raise the root to.
• The denominator tells you which root to take (e.g., square root, cube root).

In the original exercise, \( \left(-\frac{8}{27}\right)^{-\frac{1}{3}} \) directly applies the concept by indicating that you take the cube root (since \( \frac{1}{3} \) signifies a third root) of the inverted base, \( -\frac{27}{8} \). This is exactly what we do to simplify the expression into \( \frac{-3}{2} \). Understanding fractional exponents is key because they allow operations that involve not just multiplication or division of a base, but also extracting roots as easily expressed powers.