Problem 61 For Exercises $49-64$, convert... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 10: Problem 61

For Exercises $49-64$, convert to standard form. $$2.003 \times 10^{-6}$$

Short Answer

Expert verified

0.000002003

Step by step solution

- Identify the coefficient and exponent

The given notation is in scientific form: $2.003 \times 10^{-6}$. Here, 2.003 is the coefficient and $-6$ is the exponent of 10.

- Understand the meaning of the exponent

The exponent $-6$ means that the decimal point of the coefficient 2.003 should be moved 6 places to the left to convert from scientific notation to standard form.

- Move the decimal point

Shift the decimal point in 2.003 six places to the left. There aren't enough digits to the left of the existing decimal point, so add zeros as needed: Before moving: 2.003After moving: 0.000002003

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Key Concepts

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

scientific notation

Scientific notation is a method used to express very large or very small numbers in a compact form. It simplifies calculations and makes reading numbers easier.
In scientific notation, numbers are written in the form: $ a \times 10^n $. Here:

$ a $ is called the coefficient, and
$ n $ is the exponent or power of 10.

The coefficient must be a number greater than or equal to 1 and less than 10.~
For example: 2.003 x 10^{-6}~
In this example, 2.003 is the coefficient and $-6$ is the exponent.

standard form

Standard form, also known as decimal notation, is the regular way of writing numbers without exponents. In standard form, you write out the entire number without using any powers of 10.

For example, the scientific notation

2.003 x 10^{-6}

becomes 0.000002003 in standard form.
It's especially useful when you need to work with numbers in calculations or when you need a precise value instead of a simplified expression.

decimal point movement

Understanding how to move the decimal point is crucial for converting from scientific notation to standard form.

When the exponent is positive, the decimal point moves to the right.
When the exponent is negative, the decimal point moves to the left.

For the scientific notation 2.003 x 10^{-6}, we need to move the decimal point six places to the left, because the exponent is -6:

2.003 becomes 0.000002003 after moving the decimal point six places to the left.

exponent interpretation

The exponent in scientific notation tells you how many places to move the decimal point. The direction depends on whether the exponent is positive or negative.

For example:

If the exponent is 3 ($10^3$), move the decimal point 3 places to the right (e.g., 1.23 x 10^3 = 1230).
If the exponent is -4 ($10^{-4}$), move the decimal point 4 places to the left (e.g., 5.67 x 10^{-4} = 0.000567).

In our example, the exponent is -6, and it indicates that we shift the decimal point 6 places to the left, converting 2.003 x 10^{-6} to 0.000002003.

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91影视

For Exercises \(49-64\), convert to standard form. $$2.003 \times 10^{-6}$$

Short Answer

Step by step solution

- Identify the coefficient and exponent

- Understand the meaning of the exponent

- Move the decimal point

Key Concepts

scientific notation

standard form

decimal point movement

exponent interpretation

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