Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$125^{-2 / 3}$$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{25}\).

Step 1: Understand the Expression

The expression given is \(125^{-2/3}\). Our task is to evaluate this expression by simplifying it step by step.

Step 2: Simplify the Exponent

The expression \(x^{m/n}\) can be rewritten using roots and powers. Here, \(125^{-2/3}\) can be expressed as \((125^{1/3})^{-2}\). This means we are taking the cube root of 125 first, and then squaring it with a negative sign.

Step 3: Evaluate the Root

Calculate the cube root of 125. Since \(125 = 5^3\), the cube root of 125 is 5. Therefore, \(125^{1/3} = 5\).

Step 4: Apply the Negative Exponent

We now have \((125^{1/3})^{-2} = (5)^{-2}\). When we have a negative exponent, we take the reciprocal of the base raised to the positive of that exponent. So \(5^{-2} = \frac{1}{5^2}\).

Step 5: Calculate the Final Expression

Compute \(5^2\), which is 25. Therefore, \(\frac{1}{5^2} = \frac{1}{25}\). This is the final evaluated expression.

Key Concepts

Negative Exponents

In algebra, negative exponents can be tricky, but they follow simple rules that make them easy to handle. It's essential to remember that a negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For example, when we have an expression like \( a^{-n} \), it converts to \( \frac{1}{a^n} \).
This means you flip the base into the denominator with a positive exponent. Negative exponents do not mean that the value itself is negative, but rather that it's a part of a fraction.
For instance, if you encounter \( 10^{-3} \), this equals \( \frac{1}{10^3} \) or \( \frac{1}{1000} \). Using this rule simplifies calculations and helps in easily managing seemingly complex expressions.

Root Calculations

Root calculations are fundamental in simplifying expressions and understanding the nature of numbers. A common root calculation is the square root, but there are other roots, like the cube root or fourth root. In general, the \( n \)-th root of a number \( a \) is written as \( \sqrt[n]{a} \).
This operation "undoes" the process of raising a number to a power.
Consider \( \sqrt[3]{8} \), which means "what number multiplied by itself three times equals 8?" The answer is 2, because \( 2^3 = 8 \).

• Certain roots are perfect, like \( \sqrt{9} = 3 \) because \( 3^2 = 9 \).
• Understanding roots helps when working with fractional exponents, as they are closely related.

Knowing how to compute roots prepares you for more complex calculations, especially when combined with other operations like exponents.

Fractional Exponents

Fractional exponents offer a different way of expressing roots and powers in one notation. This method is often more efficient and compact in mathematical expressions. A fractional exponent like \( a^{m/n} \) represents two operations: raise \( a \) to the power of \( m \) and take the \( n \)-th root of that result.
This can be thought of as \( \sqrt[n]{a^m} \) or \( (\sqrt[n]{a})^m \).
For example, \( 16^{3/4} \) combines taking the fourth root and cubing the result:

• First, find \( \sqrt[4]{16} = 2 \), because \( 2^4 = 16 \).
• Then, raise that result to the third power: \( 2^3 = 8 \).

Working through fractional exponents requires understanding how to break down the problem into sequential steps, handling roots first and exponents second. This approach can simplify and solve complicated expressions.