Problem 81 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify the expression. $$ \left(\frac{1}{10}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is 1000.

Step 1: Understanding the exponent rule

To simplify this expression, we need to understand the negative exponent rule. The rule states that a number \(a^{-n}\) can be rewritten as \( \frac{1}{a^n} \). Thus, \( \left(\frac{1}{10}\right)^{-3} \) can be rewritten as \( \frac{1}{\left(\frac{1}{10}\right)^3} \).

Step 2: Explicit simplification

Now, we simplify \( \left(\frac{1}{10}\right)^3 \) which results in \( \frac{1}{10\times10\times10} = \frac{1}{1000} \). After this simplification, our expression is \( \frac{1}{ \left(\frac{1}{1000}\right)} \).

Step 3: Final simplification

Lastly, we simplify the final expression which is a fraction as well. The rule here is \( \frac{a}{ \left(\frac{1}{b}\right)} = a \times b \). So, we get \(1 \times 1000 = 1000 \).

Key Concepts

Negative Exponent Rule

Understanding the negative exponent rule is critical when simplifying algebraic expressions with negative powers. This rule stipulates that any base raised to a negative exponent is equal to the reciprocal of that base raised to the opposite positive exponent. In other words, the expression \( a^{-n} \) can be rewritten as \( \frac{1}{a^n} \).

Take the expression \( \left(\frac{1}{10}\right)^{-3} \) as an example. According to the negative exponent rule, we convert the negative exponent to a positive by finding the reciprocal of the base, which gives us \( \frac{1}{\left(\frac{1}{10}\right)^3} \). This step is essential because it transforms an initially confusing concept into a more familiar form, allowing for simpler multiplication or division thereafter.

Exponents and Powers

When dealing with exponents and powers, we work with repeated multiplication of a number. An exponent indicates how many times the base is multiplied by itself. For instance, \( 10^3 \) means to multiply 10 three times (\( 10 \times 10 \times 10 \) = 1,000).

In the case of fractional bases, such as \( \left(\frac{1}{10}\right)^3 \), we still follow the same process. This results in \( \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \), which simplifies to \( \frac{1}{1000} \). Applying this understanding to our original problem can significantly clarify the process of simplification and make it much more manageable.

Algebraic Expressions Simplification

The process of algebraic expressions simplification often requires a combination of multiple rules. In our original problem, after applying the negative exponent rule and understanding how to multiply powers, we arrive at the stage where we need to further simplify \( \frac{1}{\left(\frac{1}{1000}\right)} \).

To tackle this, we must recall another rule that tells us how to simplify a fraction over a fraction. This rule, often known as 'multiplying by the reciprocal,' allows us to simplify \( \frac{a}{\left(\frac{1}{b}\right)} \) into \(a \times b\). Applying this to our original expression, we convert the complex fraction into a simple multiplication: \(1 \times 1000\), leading to the final simplified answer of 1000. Mastering each of these steps – negative exponents, multiplication of powers, and simplification of complex fractions – builds a strong foundation for simplifying various algebraic expressions.