Problem 52 - Elementary and Intermediate Algebra

Question

Simplify. \(\left(\frac{1}{5}\right)^{-3}\)

Step-by-Step Solution

Answer

\(\left(\frac{1}{5}\right)^{-3} = 125\)

Step 1: Understand the Negative Exponent

Recognize that a negative exponent such as \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\). This means that \(\left(\frac{1}{5}\right)^{-3}\) can be rewritten as \(\frac{1}{\left(\frac{1}{5}\right)^3}\).

Step 2: Calculate the Positive Exponent

Convert \(\left(\frac{1}{5}\right)^3\) into a single value by raising \(\frac{1}{5}\) to the power of 3. This is calculated as: \[ \left(\frac{1}{5}\right)^3 = \frac{1^3}{5^3} = \frac{1}{125} \]

Step 3: Simplify the Expression

Now substitute \(\frac{1}{125}\) back into \(\frac{1}{\left(\frac{1}{5}\right)^3}\) to obtain: \[ \frac{1}{\left(\frac{1}{5}\right)^3} = \frac{1}{\frac{1}{125}} \]

Step 4: Calculate the Simplification

To simplify \(\frac{1}{\frac{1}{125}}\) further, recognize that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \[ \frac{1}{\frac{1}{125}} = 125 \]

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that makes complex problems more manageable and easier to solve. When we simplify expressions, we apply various mathematical rules and simplifications to express the same mathematical idea in a much simpler form.

Here are some key points about simplifying expressions:

• It usually involves combining like terms. Like terms are terms that have the same variables and powers.
• Simplification can also involve applying the distributive property, which is distributing a multiplier across terms within parentheses.
• In the case of exponents, simplifying might involve applying rules like the power of a power, product of powers, or power of a product.

These transformations help us to solve equations with fewer steps and less chance for error. In the example exercise, using the understanding of negative and positive exponents significantly simplified the problem.

Fractional Exponents

Fractional exponents are a way to express powers and roots in a single notation. For example, the square root of a number can be written as a fractional exponent. Instead of writing \(\sqrt{a}\), you can write \(a^{\frac{1}{2}}\). This notation is powerful and provides a unified method for dealing with roots and powers.

Here’s what you need to know about fractional exponents:

• A fractional exponent like \(a^{\frac{m}{n}}\) can be interpreted as \(\sqrt[n]{a^m}\), meaning raise \(a\) to the power of \(m\) and then take the \(n\)-th root.
• Fractional exponents make it simpler to perform operations involving roots, especially when dealing with higher powers or nested roots.
• They also follow the same exponent rules that we apply to integer exponents, including the power of a power and the multiplication of powers with the same base.

Using fractional exponents, the expression \(\left(\frac{1}{5}\right)^{-3}\) can be approached in a straightforward way, involving inversion and simplification.

Reciprocals

A reciprocal is simply what you multiply a number by to get 1. In more technical terms, the reciprocal of a number \(a\) (assuming \(a eq 0\)) is \(\frac{1}{a}\). Receivers feature prominently when dealing with negative exponents and fractions.

Key facts about reciprocals:

• The reciprocal operation effectively 'flips' the numerator and the denominator of a fraction. For instance, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
• Any non-zero number multiplied by its reciprocal equals 1.
• Reciprocals are especially useful for division by fractions. Dividing by a fraction equals multiplying by its reciprocal.

In the original exercise, identifying and working with reciprocals helped to turn \(\frac{1}{\left(\frac{1}{5}\right)^3}\) into a simpler form, ultimately leading to the answer of 125. Understanding reciprocals allows you to handle complex expressions with confidence and ease.