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Chapter 5: Problem 36

Write each number without exponents. $$ 7.123 \times 10^{-10} $$

Short Answer

Expert verified
0.0000000007123

Step by step solution

01

Understand the notation

The number is given in scientific notation: \(7.123 \times 10^{-10}\). This means you will need to move the decimal point.
02

Identify the direction and number of places

The exponent is \(-10\), so move the decimal point 10 places to the left.
03

Move the decimal point

Starting with 7.123, move the decimal point 10 places to the left: \(0.0000000007123\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, in the term \(10^{-10}\), the base is 10 and the exponent is -10.
The exponent can be positive or negative:
  • Positive exponents show standard multiplication (e.g., \(10^2 = 10 \times 10 = 100\)).
  • Negative exponents indicate division or how many times the number is divided by itself (e.g., \(10^{-2} = \frac{1}{10 \times 10} = \frac{1}{100}\)).
Recognizing and understanding exponents, especially negative ones, is crucial for conversions in scientific notation.
Decimal Point
The decimal point is a dot used to separate the whole number part of a number from its fractional part. For example, in the number 7.123, the 7 is the whole number, and 123 is the fractional part.

When you encounter scientific notation, you often need to move the decimal point based on the exponent:
  • If the exponent is positive, move the decimal point to the right.
  • If the exponent is negative, move it to the left.
In our example, for the number \(7.123 \times 10^{-10}\), the negative exponent means we need to move the decimal point 10 places to the left to convert it into a standard numerical form.
Scientific Notation Conversion
Scientific notation is a way of expressing very large or very small numbers in a compact form, which makes them easier to work with, especially in scientific and engineering fields.

To convert from scientific notation to standard form, follow these steps:
  • Identify the exponent. For \(7.123 \times 10^{-10}\), the exponent is -10.
  • Decide the direction to move the decimal point based on the exponent's sign. A negative exponent means moving left.
  • Move the decimal the required number of places. In this case, move 10 places left.
  • Fill in with zeros where necessary. So, \(7.123 \times 10^{-10}\) becomes 0.0000000007123.
Practice moving the decimal point both left and right to become comfortable with these conversions.

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