Problem 11 - Algebra for College Students

Question

Evaluate each numerical expression. \(\left(\frac{1}{27}\right)^{-\frac{1}{3}}\)

Step-by-Step Solution

Answer

The expression simplifies to 3.

Step 1: Understand Negative Exponents

When a number is raised to a negative exponent, it means that the base is to be reciprocated (flipped as a fraction) before applying the positive exponent. For example, if \(a^{-n}\) is given, it can be rewritten as \(\frac{1}{a^n}\). For our expression, \(\left(\frac{1}{27}\right)^{-\frac{1}{3}}\), we can write it as \(\left(27\right)^{\frac{1}{3}}\).

Step 2: Simplify the Exponent

Now, we need to evaluate \(27^{\frac{1}{3}}\). The expression \(a^{\frac{1}{3}}\) represents the cube root of \(a\).

Step 3: Calculate the Cube Root

Find the cube root of 27. Since \(27 = 3^3\), the cube root of 27 is 3. This gives us \(27^{\frac{1}{3}} = 3\).

Step 4: Conclusion

Therefore, the value of the expression \(\left(\frac{1}{27}\right)^{-\frac{1}{3}}\) simplifies to 3.

Key Concepts

Understanding Negative Exponents

Understanding negative exponents is crucial when you're dealing with mathematical expressions that may seem complex at first. A negative exponent doesn't imply the number will be negative. Instead, it denotes that you should take the reciprocal of the base, which means flipping the fraction. For instance, given an expression like \( a^{-n} \), you rewrite it as \( \frac{1}{a^n} \). This transformation helps simplify calculations by eliminating the negative sign from the exponent.

In the case of the expression \( \left(\frac{1}{27}\right)^{-\frac{1}{3}} \), you apply this concept by rewriting it to \( 27^{\frac{1}{3}} \). This step transforms the expression by turning the fraction into a whole number base with a positive exponent. It highlights the power of understanding how negative exponents function and simplifies further computations.

Reciprocal in Mathematics

The term "reciprocal" is foundational in mathematics and plays a key role in simplifying expressions with negative exponents. The reciprocal of a number is essentially its flipped version in fraction form.

For example:

• The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
• The reciprocal of 5 (which is \(\frac{5}{1}\)) is \(\frac{1}{5}\).

The reciprocal helps in transforming expressions with negative exponents into forms that are easier to work with. This process becomes straightforward when you understand that flipping the base essentially turns the negative exponent into a positive one in its reciprocal form. When simplifying \( \left( \frac{1}{27} \right)^{-\frac{1}{3}} \), recognizing that the reciprocal of \( \frac{1}{27} \) is 27 allows you to reformulate the expression into a simpler equation involving positive exponents.

Finding Cube Roots

Cube roots are another important concept when simplifying expressions and are often denoted by the exponent \( \frac{1}{3} \). The cube root of a number \( a \) is the value which, when multiplied by itself twice, gives \( a \).

Understanding how to find cube roots can simplify many expressions and equations, especially those involving powers like \( a^{\frac{1}{3}} \). For instance, if you have \( 27^{\frac{1}{3}} \), finding the cube root becomes much simpler when you know that \( 27 = 3^3 \).
This explains why the cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \). By transforming the problem into one involving cube roots, you gain clarity on how to manage expressions using this fundamental concept.

Simplifying Expressions

Simplifying expressions involves breaking down mathematical expressions into a more manageable or easily solvable form. This process often includes using various mathematical rules and concepts, such as negative exponents, reciprocals, and roots.

When simplifying expressions like \( \left( \frac{1}{27} \right)^{-\frac{1}{3}} \), you start by addressing the negative exponent by finding the reciprocal, resulting in \( 27^{\frac{1}{3}} \).
Next, you calculate the cube root, which further simplifies the expression to a single number: 3. This series of steps takes a complex expression and reduces it to its simplest form. Simplifying expressions is a valuable skill, enabling you to solve mathematical problems efficiently while honing your analytical abilities.