Problem 17 - Algebra for College Students

Question

Simplify each numerical expression. \(10^{-5} \cdot 10^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(10^{-3}\), which equals 0.001.

Step 1: Apply the Laws of Exponents

When multiplying powers with the same base, we add the exponents. In this case, the base is 10, and the exponents are -5 and 2. So, we have: \[10^{-5} imes 10^2 = 10^{-5 + 2} = 10^{-3}\]

Step 2: Simplify the Expression

The expression from Step 1 simplifies to a single power of 10: \[10^{-3}\] This represents a very small number, which is the fraction \[\frac{1}{10^3}\].

Step 3: Calculate the Decimal Value

Now, calculate the decimal value of \(10^{-3}\): \[\frac{1}{10^3} = \frac{1}{1000} = 0.001\].

Key Concepts

Exponentiation

Exponentiation is the method of raising a number, known as the base, to the power of an exponent. This is a fundamental concept in mathematics that deals with repeated multiplication of the base. For example, in the expression \(10^3\), the number 10 is the base and 3 is the exponent. This means 10 multiplied by itself 3 times: \(10 \times 10 \times 10\).

There are several important rules when working with exponents:

• Product of Powers Rule: When you're multiplying two powers with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Power of a Product Rule: Apply the exponent to each factor in the product: \((ab)^n = a^n \times b^n\).

Using these rules effectively simplifies expressions, makes calculations easier, and is particularly useful in algebra and calculus.

Negative Exponents

Negative exponents can often be a tricky concept to grasp, but they play a crucial role in mathematics. A negative exponent indicates that the base is on the opposite side of the fraction line. This might sound complicated, but it's simply following the pattern of exponents moving "down a step."

For instance, \(10^{-3}\) is the same as \(\frac{1}{10^3}\) or \(\frac{1}{1000}\). In general, any number raised to a negative exponent, \(a^{-n}\), can be rewritten as \(\frac{1}{a^n}\).

• Negative Exponents Rule: \(a^{-n} = \frac{1}{a^n}\).
• It is often used to express small numbers and aids in transforming problems to a more manageable form.
• Useful for simplifying equations and solving for variables in algebraic expressions.

Understanding negative exponents helps you handle complex mathematical expressions and is particularly useful in science and engineering where extremely small values need to be represented.

Scientific Notation

Scientific notation is a system that allows us to express very large or very small numbers in a simplified manner. This notation is widely used in science, engineering, and mathematics because it can make calculations easier and results more readable.

Scientific notation follows the format \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer. For example, the speed of light which is about 300,000,000 meters per second can be expressed as \(3 \times 10^8\) when using scientific notation.

• Usage: It allows for the easier handling, comparison, and interpretation of extreme magnitudes of numbers, such as distances in astronomy or particle size in physics.
• It helps in reducing the likelihood of errors when reading long strings of numbers.
• Simplification: Writing out long forms of numbers becomes unnecessary, keeping data neat and concise.

Learning how to convert between standard form and scientific notation is a vital skill, facilitating various calculations and analyses in scientific disciplines.