Problem 3 - Intermediate Algebra

Question

Simplify each numerical expression. \(-10^{-2}\)

Step-by-Step Solution

Answer

The simplified form of \(-10^{-2}\) is \(-0.01\).

Step 1: Understand Negative Exponents

Negative exponents indicate the reciprocal of the base with the positive exponent. In other words, if you have a term with a negative exponent, like \( a^{-n} \), it can be rewritten as \( \frac{1}{a^n} \).

Step 2: Convert Negative Exponent to Positive

Take the expression \(-10^{-2}\) and rewrite it using the concept of negative exponents. This becomes \(-\frac{1}{10^2}\) since the exponent \(-2\) becomes positive when moved to the denominator.

Step 3: Calculate the Base Raised to the Power of 2

Now simplify \(10^2\). Calculate \(10 \times 10 = 100\). Thus, \(-\frac{1}{10^2}\) becomes \(-\frac{1}{100}\).

Step 4: Simplify the Expression

The final step is understanding that the expression \(-\frac{1}{100}\) represents a negative fraction, and no further simplification is needed. The simplified version is \(-0.01\).

Key Concepts

Exponentiation

Exponentiation is a way of expressing repeated multiplication of a number by itself. In our exercise, you encountered the term \(-10^{-2}\). Here, "10" is the base, and "-2" is the exponent. Negative exponents can seem tricky at first, but they have a straightforward interpretation.

The negative sign in an exponent means that instead of multiplying, we divide. Specifically, \(a^{-n}\) is equal to \(\frac{1}{a^n}\). This means we take the reciprocal of the base raised to the absolute value of the exponent. In simple terms, to deal with a negative exponent, flip the base to the denominator and convert the exponent to positive.

• For example, \(10^{-2}\) becomes \(\frac{1}{10^2}\).
• This property allows us to express equations without a negative exponent, making them easier to handle.

Fraction Simplification

Simplifying fractions involves rewriting them in their simplest form. In our solved problem, we converted a negative exponent to a positive fraction: \(-\frac{1}{10^2}\).

To simplify the fraction, you calculate the power of the base. For \(10^2\), multiply 10 by itself, which results in 100. The expression \(-\frac{1}{10^2}\) therefore simplifies to \(-\frac{1}{100}\). This fraction represents a very small negative number.

• When simplifying fractions, ensure that the numerator and denominator are the smallest possible numbers representing the fraction.
• Sometimes, simplification may involve dividing both the numerator and denominator by their greatest common divisor, but in our example, the fraction \(-\frac{1}{100}\) is already in its simplest form.

Basic Arithmetic Concepts

Understanding basic arithmetic concepts helps simplify expressions like \(-10^{-2}\). The core elements of arithmetic include addition, subtraction, multiplication, and division. In many cases, breaking down an expression into smaller, more manageable parts can lead to a simpler overall solution.

In this exercise, calculating \(10^2\) involved a basic multiplication operation: \(10\times10 = 100\). Knowing this multiplication step is crucial as it allows us to simplify other arithmetic expressions involving powers.

• Basic arithmetic is not just about computation; it’s also about understanding the principles that guide each operation.
• Becoming comfortable with these operations enables you to tackle more complex expressions, like those involving negative exponents, more intuitively.