Problem 11 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of the expression is 3.

Step 1: Simplify the Expression

First, understand the expression \( \left(\frac{1}{27}\right)^{-\frac{1}{3}} \). The negative exponent indicates that we will take the reciprocal of the base \( \frac{1}{27} \) and then raise it to the positive exponent \( \frac{1}{3} \).

Step 2: Apply the Negative Exponent

Applying the negative exponent rule, we take the reciprocal of \( \frac{1}{27} \), which gives us \( 27 \). Now, the expression becomes \( 27^{\frac{1}{3}} \).

Step 3: Evaluate the Cube Root

The exponent \( \frac{1}{3} \) represents the cube root. Therefore, we need to find \( 27^{\frac{1}{3}} \), which is \( \sqrt[3]{27} \). This evaluates to 3, since \( 3^3 = 27 \).

Key Concepts

Negative Exponents

Understanding negative exponents is a crucial algebraic concept. It primarily deals with reciprocals, or flipping the base of the exponent. When you see an expression like \( a^{-n} \), this simply means you take the reciprocal of the base \( a \), and raise it to the positive of the given power. In mathematical terms, \( a^{-n} = \frac{1}{a^n} \).

Here are a few key points about negative exponents:

• It does not mean the result is negative, just that you need to flip the fraction.
• Negative exponents are a shortcut to writing expressions that involve reciprocals.
• They can be applied to whole numbers, fractions, and decimals.

So, in the expression \( \left(\frac{1}{27}\right)^{-\frac{1}{3}} \), the negative exponent \(-\frac{1}{3}\) means we take the reciprocal of \( \frac{1}{27} \), which is \( 27 \), and then deal with any remaining positive exponents.

Reciprocal

Taking the reciprocal is a friendly operation you'll often meet in algebra, especially when dealing with negative exponents. But what really is a reciprocal? When you take the reciprocal of a number, you essentially flip its fraction. That is, \( a \) becomes \( \frac{1}{a} \). Similarly, for a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).

Here’s what happens with reciprocals:

• The product of a number and its reciprocal is always 1. For example, \( 3 \times \frac{1}{3} = 1 \).
• Reciprocals are really useful for dividing fractions, since dividing is the same as multiplying by the reciprocal.
• They help transform negative exponents into something more manageable by giving us the base to a positive exponent.

In our expression, when we apply the reciprocal to \( \frac{1}{27} \), we convert it into \( 27 \), which is much easier to handle, especially with cube roots.

Cube Root

The cube root is a wonderful tool in algebra, symbolized by the power of \( \frac{1}{3} \). It asks you, "What number multiplied by itself three times gives me the original number?" In mathematical terms, you find \( \sqrt[3]{x} \) which is \( x^{\frac{1}{3}} \).

Key insights about cube roots:

• The cube root of a perfect cube returns to a neat whole number. Such as \( \sqrt[3]{27} = 3 \).
• Cube roots can be taken of both positive and negative numbers, unlike the square root, because a negative number cubed remains negative.
• Cube root operations can help simplify expressions or change the form of a problem to make it easier to solve.

In our exercise, after taking the reciprocal of \( \frac{1}{27} \) to get \( 27 \), we need to find \( \sqrt[3]{27} \), which evaluates to 3. This confirms that 3 multiplied by itself thrice gives us 27.