Problem 40 - Elementary Algebra

Question

\(\left(6 \times 10^{-4}\right)^{2}\)

Step-by-Step Solution

Answer

36 \times 10^{-8}

Step 1 - Understand the Expression

The given expression is \((6 \times 10^{-4})^2\). It is a number in scientific notation raised to the power of 2.

Step 2 - Separate Components

Separate the base (6) and the power of 10 \((10^{-4})\). Rewrite the expression as \((6^2 \times (10^{-4})^2)\).

Step 3 - Square the Base

Calculate \(6^2\). \(6^2 = 36\).

Step 4 - Square the Power of 10

Calculate \((10^{-4})^2\). Use the property of exponents that \((a^m)^n = a^{m \times n}\). Thus, \((10^{-4})^2 = 10^{-8}\).

Step 5 - Combine the Results

Multiply the results from Steps 3 and 4. So, \(6^2 \times (10^{-4})^2 = 36 \times 10^{-8}\).

Step 6 - Write the Final Answer

The final answer is \(36 \times 10^{-8}\).

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication of the same number. For example, in the expression \(3^4\), the number 3 is multiplied by itself 4 times, resulting in 81. Mathematically, this is represented as: \[3^4 = 3 \times 3 \times 3 \times 3 = 81\].
Exponents have properties that make calculations easier:

• \((a^m)^n = a^{m \times n}\)
• \(a^m \times a^n = a^{m+n}\)
• If \(a eq 0\), then \(a^0 = 1\)
• \(a^{-m} = \frac{1}{a^m}\)

Understanding these properties is crucial when dealing with scientific notation and other mathematical operations involving exponents.

Base and Power

In expressions involving exponents, the base is the number that is being multiplied, and the exponent (or power) indicates how many times the base is multiplied by itself.
In the expression \(6 \times 10^{-4}\), the base 6 is expressed in a standard format, while \(10^{-4}\) is the base with an exponent. Here, \(10\) is the base and \(-4\) is the exponent, signifying that the base \(10\) should be divided four times (or multiplied by its reciprocal four times): \[10^{-4} = \frac{1}{10^4} = \frac{1}{10000} = 0.0001\]
When working with scientific notation, it's important to understand how to separate and manipulate the base and power to simplify expressions effectively.

Multiplication of Powers

Multiplying powers involves combining like terms and using the properties of exponents. In the given exercise, we see an application of this concept: \[ (6 \times 10^{-4})^2 \]
The steps involve separating the base and the power of 10 and then applying the exponent to each part:

• First, separate: \( (6^2) \times (10^{-4})^2 \)
• Square the base: \( 6^2 = 36 \)
• Apply the exponent to the power of 10: \( (10^{-4})^2 = 10^{-8} \)
• Combine the results: \( 36 \times 10^{-8} \)

The final answer is \( 36 \times 10^{-8} \). Understanding how to multiply powers helps in simplifying complex expressions and solving equations efficiently.