Problem 12 - Algebra 2 and Trigonometry

Question

Write each number as a power. 0.125

Step-by-Step Solution

Answer

0.125 can be expressed as \(2^{-3}\).

Step 1: Understand Powers and Exponents

In mathematics, a power represents the multiplication of a number by itself a certain number of times. It is written as a base number with an exponent. For example, if we have 2 raised to the power of 3, it is written as \(2^3\), which equals 2 \(\times 2 \times 2 = 8\). Our task here is to express 0.125 as a power.

Step 2: Recognize a Common Base

Observe that 0.125 is a fractional number. A common base in power form often used to express fractions like 0.125 is the base 2, because fractions like 0.5 (\(2^{-1}\)), 0.25 (\(2^{-2}\)), and 0.125 (\(2^{-3}\)) are powers of 2. Let's express 0.125 as a power of 2.

Step 3: Convert 0.125 to a Fraction

0.125 is equivalent to the fraction \(\frac{1}{8}\). This fraction is important to work with because it can be rewritten using a power.

Step 4: Rewrite the Fraction Using a Power

Recognizing that \(8\) can be rewritten as \(2^3\), the fraction \(\frac{1}{8}\) can be represented as \(\frac{1}{2^3}\).

Step 5: Apply Negative Exponents

A fraction like \(\frac{1}{2^3}\) can be rewritten using a negative power as \(2^{-3}\). This is the final power representation of 0.125.

Key Concepts

Negative Exponents

Understanding negative exponents can initially seem challenging, but it's an essential part of working with powers. When we talk about exponents, we're typically thinking about positive exponents like in the expression \(3^2 = 9\). But exponents can also be negative, which changes how we view the multiplication.

• A negative exponent indicates a reciprocal. For example, \(a^{-n} = \frac{1}{a^n}\). Here, instead of multiplying \(a\) by itself \(n\) times, you're taking the reciprocal of \(a\) multiplied by itself \(n\) times.
• Using negative exponents makes the expression of fractions in powers easier to handle and comprehend. So \(\frac{1}{8} = \frac{1}{2^3}\) can be represented as \(2^{-3}\).

Understanding how to utilize negative exponents lets you switch between fractions and powers seamlessly, which simplifies many mathematical operations.

Fraction to Power Conversion

Converting fractions to powers is a nifty math trick that helps in simplifying calculations. This process often involves seeing fractions as the result of division involving powers of whole numbers. Consider the fraction \(\frac{1}{8}\).

• To convert it to a power form, you need to identify the base number that, when multiplied by itself a certain number of times, gives you the denominator. Here, \(8\) can be expressed as \(2^3\), since multiplying \(2\) by itself three times gives \(8\).
• With this understanding, \(\frac{1}{8}\) can be converted to \(\frac{1}{2^3}\). Applying the rules of exponents, it further simplifies to \(2^{-3}\).

This conversion makes it possible to move between fractions and exponential forms, proving extremely useful in equations where simplifying terms is necessary.

Base Numbers in Powers

The base number in a power expression tells us which number is being repeatedly multiplied. Recognizing base numbers is a key skill in working effectively with powers and exponents. Let’s take a closer look:

• In an expression like \(2^3\), \(2\) is the base number, while \(3\) is the exponent. It means that \(2\) is multiplied by itself a total of 3 times: \(2 \times 2 \times 2 = 8\).
• Choosing the correct base number is crucial, especially when converting fractions or decimals to power form. It requires recognizing factors that can easily represent numbers like decimals, such as turning \(0.125\) into \(2^{-3}\) by recognizing that \(8\) (the denominator of \(\frac{1}{8}\)) is \(2^3\).

Adequately understanding base numbers aids in making complex expressions simpler and helps in solving problems with ease.