Problem 24 - Elementary Algebra

Question

Find the value of \(\left(4 \times 10^{-16}\right)^{2}\)

Step-by-Step Solution

Answer

Answer: \(16 \times 10^{-32}\)

Step 1: Multiply 4 by \(10^{-16}\)

To find the product of \(4 \times 10^{-16}\), we simply multiply the base numbers together and retain the exponent of the power of 10. In this case, since we're multiplying 4 by 1, the base number remains 4. The exponent is \(-16\), so the result is \(4 \times 10^{-16}\).

Step 2: Square the result

Now, we need to square the result obtained in step 1. To square a number, we multiply it by itself. So, the square of \(4 \times 10^{-16}\) will be: \((4 \times 10^{-16})^2 = (4 \times 10^{-16}) \times (4 \times 10^{-16})\) In order to multiply these two numbers together, we will first multiply the base numbers: \(4 \times 4 = 16\). Next, we will add the exponents of the powers of 10. Since both exponents are -16, the sum is \(-16 + (-16) = -32\). The result is: \((4 \times 10^{-16})^2 = 16 \times 10^{-32}\) So, the value of \((4 \times 10^{-16})^2\) is \(16 \times 10^{-32}\).

Key Concepts

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a compact form. It is especially useful in science and mathematics. This system simplifies such numbers by expressing them as a product of a coefficient and a power of 10.

In scientific notation, a number is written as:

• A coefficient (a number between 1 and 10)
• Followed by a multiplication sign
• And a power of 10

For example, the number 4 in scientific notation is written as \( 4 \times 10^{0} \) because \( 10^{0} = 1 \). Likewise, \( 4 \times 10^{-16} \) is a scaled-down version of 4 shifted 16 decimal places to the left, denoting a very small number.

Multiplying Powers

When multiplying numbers in scientific notation, we can separately deal with the coefficients and the exponents of 10. This makes calculations simpler and more intuitive.

To multiply powers with the same base, follow these steps:

• Multiply the coefficients together as usual. For example, multiplying 4 by 4 gives 16.
• Add the exponents of the base 10 using exponent rules. In our exercise, \( -16 + (-16) = -32 \).

Using these steps allows for an easy multiplication of numbers in scientific notation, resulting in an efficient way to handle both large and small quantities.

Exponent Rules

Exponent rules are a set of guidelines used to simplify algebraic expressions involving powers. Here are some key rules concerning multiplying powers:

• Product of Powers Rule: When you multiply two powers with the same base, simply add their exponents. For instance, \( a^{m} \times a^{n} = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, \((a^{m})^{n} = a^{m \cdot n} \).
• Power of a Product Rule: To raise a product to a power, raise each factor to the power \((ab)^{n} = a^{n} b^{n} \).

Understanding these rules helps simplify complex equations, making it easier to solve problems involving powers.

Mathematics Education

Understanding the concept of exponents and scientific notation is crucial in mathematics education as it lays a foundational skill for solving complex real-world problems. Efficiently grasping these concepts enables students to:

• Work with very large and very small numbers, which is crucial in fields like physics, chemistry, and engineering.
• Simplify calculations that would otherwise be cumbersome and time-consuming.
• Enhance mathematical problem-solving abilities, contributing to stronger analytical skills.

Integrating technology and interactive learning methods can further simplify these concepts, thereby enriching the educational experience.