Problem 9 - Algebra 2 and Trigonometry

Question

Write each number as a power. \(\frac{1}{8}\)

Step-by-Step Solution

Answer

\(\frac{1}{8}\) is \(2^{-3}\).

Step 1: Understand the Problem

We need to express \(\frac{1}{8}\) as a power, meaning we want to write it in the form of a base raised to an exponent.

Step 2: Convert to a Fraction in Base 2

Recognize that 8 is a power of 2. Specifically, \(8 = 2^3\). Thus, \(\frac{1}{8}\) can be written as \(\frac{1}{2^3}\).

Step 3: Apply Negative Exponent Rule

The expression \(\frac{1}{2^3}\) can be rewritten using the negative exponent rule: \(\frac{1}{a^n} = a^{-n}\). Therefore, \(\frac{1}{2^3}\) becomes \(2^{-3}\).

Step 4: Final Expression as a Power

Conclude that \(\frac{1}{8}\) can be written as the power \(2^{-3}\).

Key Concepts

fraction to power conversion

When we talk about "fraction to power conversion," we aim to express a fraction as an exponentiation involving a base and an exponent. This process is particularly useful in simplifying expressions and understanding the underlying mathematical structure. For example, to convert the fraction \( \frac{1}{8} \) into a power, we must first recognize patterns related to exponents.

The number 8 can be expressed as a power of 2, since \( 8 = 2^3 \). Therefore, the fraction \( \frac{1}{8} \) is equivalent to \( \frac{1}{2^3} \). Recognizing this equivalence is a crucial step in the conversion process.

By identifying such patterns, we can translate fractions into a more manageable exponential form, which allows us to use rules of exponents for further simplifications or evaluations without changing the value of the expression.

base and exponent

In mathematics, a number expressed as a power consists of a base and an exponent. Understanding the roles of the base and exponent is fundamental to mastering exponents.

The "base" is the number that is being multiplied by itself a certain number of times. For example, in the expression \(2^3\), 2 is the base. It is the number that will be used repeatedly in the multiplication. This is like saying "multiply 2 by itself."

• The "exponent" or "power" is the small number written above and to the right of the base. It indicates how many times to multiply the base by itself. In our example, \(2^3\), the exponent is 3, showing that the base should be used in multiplication three times such that it's equivalent to \(2 \times 2 \times 2\).

Exponents help in organizing repeated multiplication into concise forms. Knowing how to interpret these components plays a vital role in solving exponent-related problems.

negative exponent rule

The negative exponent rule is a powerful tool when dealing with fractions and exponents. According to this rule, if you have a number in the denominator of a fraction represented as a power, you can convert it into a negative exponent in the numerator. The rule states: \( \frac{1}{a^n} = a^{-n} \).

Understanding this rule allows us to transform fractions into a simpler exponential form and vice versa. For example, when you see \( \frac{1}{2^3} \), using the negative exponent rule, it becomes \(2^{-3}\).

This transformation is not just a mathematical trick. It enables greater flexibility in solving equations, simplifies expressions, and is commonly used in calculus and higher mathematics. By mastering the negative exponent rule, you expand your toolkit for handling a variety of math challenges efficiently.