Problem 4 - Algebra for College Students

Question

Simplify each numerical expression. \(10^{-3}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{1}{1000}\).

Step 1: Understand Negative Exponent Rule

The negative exponent rule tells us that a number raised to a negative exponent is the same as one over that number raised to the opposite positive exponent. This can be expressed as: for any non-zero number \(a\), \(a^{-n} = \frac{1}{a^n}\).

Step 2: Apply the Rule to Simplify the Expression

Using the negative exponent rule, we rewrite \(10^{-3}\) as \(\frac{1}{10^3}\). This replaces the negative exponent with a positive exponent in the denominator.

Step 3: Calculate the Value

Calculate \(10^3\), which means multiplying 10 by itself three times: \(10 \times 10 \times 10 = 1000\).

Step 4: Substitute Back into the Fraction

Substitute the value from Step 3 back into the fraction: \(\frac{1}{10^3} = \frac{1}{1000}\). This gives the simplified form of the expression.

Key Concepts

Exponent Rules

When working with exponents, you have several important rules that dictate how to handle them. One key rule is the **negative exponent rule**. This rule states that any non-zero number raised to a negative exponent can be expressed as the reciprocal of the number raised to the corresponding positive exponent. Put simply:

• If you have a number like \(a^{-n}\), you can transform it into \(\frac{1}{a^n}\).
• This is handy because it turns a negative exponent into something much more manageable: a positive one.

Others might include the product of powers rule or the power of a power rule, which cover different scenarios when dealing with exponents. For our task here, though, the negative exponent rule is what we need.
Understanding and applying these rules are crucial steps when simplifying expressions that involve exponents.

Numerical Expressions

Numerical expressions are essentially calculations involving numbers and operations. You can find these expressions in various forms, like addition, subtraction, multiplication, division, and of course, exponents.

• Expressions with exponents simplify numbers raised to a power, representing repeated multiplication.
• For example, in the expression \(10^{-3}\), "\(10\)" is the base, and "\(-3\)" is the exponent.

Expressions like these are an important component of algebra and enable you to handle calculations succinctly and efficiently. By understanding how to manipulate these expressions using exponent rules, you can convert seemingly complex expressions into simpler forms. This not only makes calculations easier but also helps to clearly communicate mathematical ideas.

Simplifying Expressions

The goal of simplifying expressions is to make them as simple as possible while retaining their value. For expressions with exponents, you follow particular steps:

• Use exponent rules (like the negative exponent rule) to rewrite the expression. For \(10^{-3}\), you rewrite it as \(\frac{1}{10^3}\).
• Calculate the value of any exponents. Here, \(10^3\) means multiplying \(10\) by itself three times, which equals \(1000\).
• Substitute back into the expression to finalize it. So, you end up with \(\frac{1}{1000}\).

Through this process, a complex expression with a negative exponent becomes a straightforward fraction. This approach not only reduces complexity but also aids in achieving accuracy in calculations. Simplifying expressions is a vital skill in mathematics, and mastering it can greatly enhance problem-solving efficiency.