Problem 70 - Elementary Algebra

Question

Perform the following operations. $$ \left(1.1 \times 10^{6}\right)^{2} $$

Step-by-Step Solution

Answer

Answer: The result of squaring the number \(1.1 \times 10^{6}\) is \(1.21 \times 10^{12}\).

Step 1: Square the base number

To square the base number, multiply 1.1 by itself which is \((1.1)^2\). $$ (1.1)^2 = 1.1 \times 1.1 = 1.21 $$

Step 2: Square the power of 10

To square the power of 10 part, apply the exponent rules which involve multiplying exponents with the same base. When you square a number with an exponent, you simply multiply the exponent by 2. In this case, the exponent is 6, so we multiply 6 by 2 to get the new exponent. $$ \left(10^{6}\right)^{2} = 10^{6 \times 2} = 10^{12} $$

Step 3: Combine the results

Now, combine the results of Step 1 and Step 2 to find the answer. $$ \left(1.1 \times 10^{6}\right)^{2} = (1.21)\times(10^{12}) $$ So, the final result is \(1.21 \times 10^{12}\).

Key Concepts

Understanding Exponent Rules

Exponent rules are like magic tricks for numbers that help simplify complex operations. When you have an exponent, it tells you how many times to multiply the base number by itself.
There are specific rules you need to remember for working with exponents:

• Product of Powers Rule: When you multiply numbers with the same base, simply add their exponents, like this: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When you raise an exponent to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Power of a Product Rule: When you take the power of a product, distribute the exponent to each factor: \((ab)^n = a^n \times b^n\).

In the exercise given, we used the Power of a Power Rule. By squaring the power of 10, we multiplied the exponent 6 by 2 to get 12. This gives \((10^6)^2 = 10^{12}\). Understanding these rules helps perform calculations efficiently and correctly.

Squaring Numbers Simplified

Squaring numbers simply means multiplying a number by itself. It's a straightforward process that can sometimes yield surprising results! For instance, in the exercise, we squared 1.1:
\[(1.1)^2 = 1.1 \times 1.1 = 1.21\]
Squaring is useful in various mathematical scenarios:

• Geometry: Squaring numbers is often used when calculating areas of squares, since area is side squared.
• Algebra: Squaring can help solve quadratic equations where terms appear as \(x^2\).
• Mathematical Patterns: Squaring numbers can help discover patterns, like perfect squares such as 1, 4, 9, 16, etc.

Understanding how to square numbers is essential for solving many mathematical problems, as it relates to basic concepts of multiplication and repeated addition.

Multiplying Powers of Ten

Multiplying powers of ten is really about managing exponents, especially when dealing with scientific notation. This is a powerful way to handle very large or very small numbers efficiently, such as the massive result from the original exercise of \(1.21 \times 10^{12}\).
When you multiply powers of ten, the exponents do the talking:

• Basic Concept: Multiply the bases (if necessary) and then use the product of powers rule: \(10^m \times 10^n = 10^{m+n}\).
• No Adjustment Needed: If you are simply multiplying to increase complexity or steps as in \((10^6)^2\), apply the power of a power rule as seen earlier.

This principle of working with powers of ten is critical when working with scientific notation, making it easier to read, understand, and compare extremely large or tiny numbers.