Problem 24 - Algebra 1

Question

DECIMAL FORM Rewrite in decimal form. $$ 1.1098 \times 10^{10} $$

Step-by-Step Solution

Answer

The decimal form of \(1.1098 \times 10^{10}\) is 11098000000.

Step 1: Understand the Scientific Notation

We have a number in scientific notation as \(1.1098 \times 10^{10}\). Here, 10 is raised to the power of 10 (10^10), which in scientific notation means moving the decimal point 10 places to the right.

Step 2: Moving the Decimal Point

Move the decimal 10 places to the right in the number 1.1098. When a decimal is moved to the right, additional zeroes are added in the spaces created. In this case, after the first 4 places, there would be 6 more steps to reach 10. Fill these with zeros.

Step 3: Write the Decimal Form

After moving the decimal 10 places to the right and filling with zeros, we get the decimal form as 11098000000.

Key Concepts

Decimal Form

To rewrite a number in decimal form, you follow a straightforward process. Decimal form denotes a number as we typically see it, without any exponents or scientific notation. As was initially given, in scientific notation, the number needs its decimal point moved according to a specific power of ten. Here, it's important to maintain the integrity of the number by accurately placing any necessary zeros.

• This conversion process enables easier comprehension and use, especially in everyday calculations.
• For instance, converting from something like scientific notation, we manipulate the number to its full numeral likeness.
• The nature of decimal places and movement depends directly on the power of ten, whether the decimal moves to the right or left of its original position.

This makes decimal form a very accessible and convenient way of expressing large or small numbers in mathematical and real-world contexts.

Exponents

Exponents are a way of expressing repeated multiplication of the same number. In the expression for scientific notation, such as in the given exercise, exponents simplify large-scale calculations or representations of numbers.

• In mathematical terms, an exponent indicates how many times the base (here, 10) is used as a factor.
• For example, in the expression \(10^{10}\) from the exercise, 10 is multiplied by itself 10 times.
• These concepts drastically shorten larger numbers, making them easier to manipulate or understand initially.

Understanding exponents is critical as they are foundational in scientific notation and many areas of mathematics and science education.

Powers of Ten

Powers of Ten are vital in reshaping numbers, especially in scientific notation. This powerful concept involves raising the number 10 by any exponent, dwelling on the essence of moving the decimal point in either direction.

• This is instrumental in simplifying very large or very small numbers, enabling effortless handling.
• Each increase in the power signifies more zeros needed when transforming the number to decimal form.
• In our exercise, \(10^{10}\) implies moving the decimal point 10 places right, effectively expanding the number's scale.

Powers of Ten are ubiquitous in fields like astronomy, electron-scale physics, or when quantifying distance in terms of light-years or microscopic dimensions. Learning to work with and understand these powers helps appreciate the vast or minuteness of quantities within scientific contexts.