Problem 7 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$125^{-2 / 3}$$

Step-by-Step Solution

Answer

\(125^{-2/3} = \frac{1}{25}\).

Step 1: Understand the Expression

The expression given is \(125^{-2/3}\). This is a power expression that involves a base (125) and an exponent (-2/3). The negative exponent indicates a reciprocal, and the fractional exponent represents a root.

Step 2: Convert to Reciprocal

Since the exponent is negative, we convert \(125^{-2/3}\) to its reciprocal form. This means \(125^{-2/3} = \frac{1}{125^{2/3}}\). Now, we only need to handle \(125^{2/3}\).

Step 3: Break Down the Fractional Exponent

The exponent \(2/3\) indicates that you should first take the cube root and then square the result. This can be expressed as \((125^{1/3})^2\).

Step 4: Calculate the Cube Root

Find the cube root of 125. Since \(125 = 5^3\), we have \(125^{1/3} = 5\).

Step 5: Square the Result

Now, take the result from the cube root and square it: \(5^2 = 25\). So \(125^{2/3} = 25\).

Step 6: Final Calculation

Since \(125^{-2/3} = \frac{1}{125^{2/3}}\) and we found \(125^{2/3} = 25\), then \(125^{-2/3} = \frac{1}{25}\).

Key Concepts

Negative Exponents

A negative exponent in mathematical expressions often confuses students at first glance. However, understanding the concept is simpler when you break it down step-by-step. When an exponent is negative, it doesn’t mean a negative number but rather a reciprocal. For example, if you have a number like \( a^{-b} \), it is equivalent to \( \frac{1}{a^b} \).

This means you take the base number \( a \) and raise it to the power of \( b \), then place it under a 1 as a denominator to find the reciprocal.

• Negative exponents represent reciprocal operations.
• Turn \( a^{-b} \) into \( \frac{1}{a^b} \).
• Think of flipping the base from the top (numerator) to the bottom (denominator).

So, when you have the expression \( 125^{-2/3} \), as in the original exercise, it turns into \( \frac{1}{125^{2/3}} \). From there, the task is to simply calculate the positive power of the reciprocal.

Fractional Exponents

Fractional exponents can seem tricky, but they have a logical breakdown that makes them easier to understand. A fractional exponent like \( a^{m/n} \) represents two operations: a root and a power.

The denominator \( n \) indicates the root, while the numerator \( m \) indicates the power.

• Fractional exponents combine roots and powers.
• \( a^{m/n} \) corresponds to \( (\sqrt[n]{a})^m \).
• Always take the root first, then raise to the power.

For the problem \( 125^{2/3} \), it translates to taking the cube root of 125 first (denominator), which is 5, then raising that result to the power of 2 (numerator), resulting in \( 5^2 = 25 \). This approach makes fractional exponents less daunting.

Cube Roots

When dealing with cube roots, you're solving for a number which, when multiplied by itself three times, returns the original number. The cube root notation is \( \sqrt[3]{a} \), and with exponents, it's written as \( a^{1/3} \).

The cube root specifically highlighted here was \( 125^{1/3} \). You need to identify whether a number is a perfect cube, meaning it can be expressed as another integer raised to the third power.

• Cube roots are about finding the number which gives the original value when cubed.
• \( \sqrt[3]{125} = 5 \) because \( 5^3 = 125 \).
• This operation helps in breaking down fractional exponents.

Finding cube roots directly helps simplify expressions involving fractional exponents, just as seen in the exercise where you deduced that \( 125^{1/3} = 5 \) before proceeding with other calculations.