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Chapter 7: Problem 40

Convert each number into standard notation. $$1.76 \times 10^{-5}$$

Short Answer

Expert verified
0.0000176

Step by step solution

01

Understand Scientific Notation

Scientific notation is a way of expressing very large or very small numbers. It is written as the product of a number between 1 and 10 and an integer power of 10. In this case, the given number is written as $$1.76 \times 10^{-5}$$.
02

Identify Components

The number provided has two parts: the coefficient (1.76) and the exponent (-5). The exponent indicates how many places to move the decimal point. A negative exponent means moving the decimal to the left.
03

Move the Decimal Point

Starting with the coefficient 1.76, move the decimal point 5 places to the left because the exponent is -5. Adding extra zeros as needed, 1.76 becomes 0.0000176.

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Key Concepts

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

Standard Notation
Standard notation is the regular way of writing numbers without exponents. It鈥檚 the familiar format that we use in everyday life. For instance, instead of using scientific notation like \(1.76 \times 10^{-5}\), we convert it into a standard form where the decimal is placed correctly.
Moving the decimal point is key to this conversion. Here鈥檚 how to change a number from scientific notation to standard notation:
  • Understand the exponent part of the scientific notation.
  • Determine the direction to move the decimal point: left for negative exponents and right for positive ones.
  • Move the decimal point and adjust the number by filling in zeros if required.
In the example \(1.76 \times 10^{-5}\), the exponent is -5, so we move the decimal point 5 places to the left, giving us \0.0000176.\
Exponents
Exponents are a shorthand way of expressing repeated multiplication of the same number. For example, \(10^2\) means \(10 \times 10 = 100\). They have a crucial role in scientific notation for representing very large or very small numbers. The structure of any number in scientific notation is \(a \times 10^n\):
  • \(a\) is the coefficient and should be between 1 and 10.
  • \(10^n\) dictates how many times to multiply \a\ by 10.
In the exercise, \(1.76 \times 10^{-5}\), \1.76\ is the coefficient, and the \-5\ is the exponent, telling us to divide \1.76\ by \100000\ (or \(10^5\)).
Decimal Point Movement
Moving the decimal point is the key action in converting from scientific to standard notation. The direction and number of places depend on the exponent:
  • A positive exponent means moving the decimal point to the right.
  • A negative exponent means moving it to the left.
For the number \(1.76 \times 10^{-5}\), the \-5\ exponent tells us to move the decimal point 5 places to the left. Starting at \1.76\, we move the decimal left:
  • After 1 move: \0.176\.
  • After 2 moves: \0.0176\.
  • After 3 moves: \0.00176\.
  • After 4 moves: \0.000176\.
  • After 5 moves: \0.0000176\.
So, \(1.76 \times 10^{-5}\) in standard notation becomes \0.0000176.\

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