Problem 54 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation without using a calculator. Write the result in scientific notation. $$ \left(2 \times 10^{-3}\right)^{4} $$

Step-by-Step Solution

Answer

The result is \(1.6 \times 10^{-11}\).

Step 1: Distribute the Power

According to the rules of exponents, when raising a product to a power, you distribute the power to each part. Thus, we will have \( (2^4) \times (10^{-3})^4 \).

Step 2: Apply the Exponent to Each Part

Calculate the 4th power of 2 and the 4th power of \(10^{-3}\). This will give us \( 16 \times 10^{-12} \).

Step 3: Write in Proper Scientific Notation

The number 16 is not in the correct form for the mantissa in a scientific notation. It should be between 1 and 10. Therefore, rewrite 16 as \(1.6 \times 10^1\). Substituting this back into our result from Step 2, we get \( (1.6 \times 10^1) \times 10^{-12} \).

Step 4: Multiply the Powers of Ten

When multiplying powers of ten, we add the exponents according to the laws of exponents. Therefore, the product of \(10^1\) and \(10^{-12}\) is \(10^{-11}\). So, the result from Step 3 expressed in scientific notation is \(1.6 \times 10^{-11}\).

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. If you see a number with an exponent, like in the expression \(2^4\), it means you should multiply 2 by itself 4 times, resulting in \(2 \times 2 \times 2 \times 2 = 16\). The number 2 is called the base, and 4 is the exponent. Exponents make it easier to write and work with very large or very small numbers.

When dealing with exponents, it helps to keep a few things in mind:

• If the exponent is zero, the result is always 1. For example, \(10^0 = 1\).
• If the exponent is 1, the result is the base itself, like in \(5^1 = 5\).
• Negative exponents represent division or a fraction, such as \(10^{-1} = \frac{1}{10}\).

Understanding these basics is crucial, as they form the foundation for more complex operations involving exponents.

Exploring the Laws of Exponents

The laws of exponents provide a set of rules that make it easier to handle calculations involving powers. Knowing these laws can help you simplify expressions efficiently.

Here are some important laws of exponents:

• Product of Powers: When multiplying two powers that have the same base, you add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another power, multiply the exponents. For instance, \((a^m)^n = a^{m \times n}\).
• Power of a Product: When raising a product to an exponent, distribute the exponent to each term. As in our exercise, \((ab)^m = a^m \times b^m\).

These laws simplify many calculations and are widely used in mathematics, especially when simplifying expressions for easier computation or derivation.

Multiplying Powers of Ten

When working with scientific notation, multiplying powers of ten is a frequent task. Powers of ten have very straightforward rules thanks to the laws of exponents. For example, multiplying \(10^1\) by \(10^{-12}\) involves adding the exponents, which gives us \(10^{1 + (-12)} = 10^{-11}\).

Here are some steps to consider while multiplying powers of ten:

• Add the exponents if the bases are the same.
• Keep an eye on positive and negative signs of the exponents. These indicate whether your number is increasing or decreasing in order of magnitude.
• Use scientific notation to simplify and express very large or very small numbers easily.

Understanding how to precisely handle multiplication of powers of ten will streamline working with scientific notation and assist in communicating large scale values effectively.

Raising a Product to a Power

Raising a product to a power involves a specific rule from the laws of exponents. This means that when you have an expression like \((ab)^m\), you distribute the exponent \(m\) to both \(a\) and \(b\), resulting in \(a^m \times b^m\).

Let's break it down in the context of our example:

• Consider \((2 \times 10^{-3})^4\). According to the rule, this becomes \(2^4 \times (10^{-3})^4\).
• Compute each part separately: \(2^4 = 16\) and \((10^{-3})^4 = 10^{-12}\).
• Combine the results: \(16 \times 10^{-12}\).

This rule is especially useful when dealing with expressions in scientific notation. It helps in breaking down complex expressions into manageable parts, simplifying the computation.