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Chapter 4: Problem 28

Write each number in standard notation. $$ -7.68 \times 10^{-4} $$

Short Answer

Expert verified
-0.000768

Step by step solution

01

Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers. The general form is: \[ a \times 10^n \] where \(a\) is a number between 1 and 10, and \(n\) is an integer exponent.
02

Identify Components

In the given number \[ -7.68 \times 10^{-4} \], \(-7.68\) is the coefficient \(a\), and \(-4\) is the exponent \(n\).
03

Interpret the Exponent

The exponent \(-4\) means that the decimal point in \(-7.68\) should be moved 4 places to the left. When the exponent is negative, the number becomes smaller.
04

Move the Decimal Point

Start with \(-7.68\). Move the decimal point 4 places to the left to get: \[ -0.000768 \]

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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 a way of writing numbers without exponents. It allows us to represent numbers in a more commonly understood manner, without using scientific notation.
For example, the scientific notation \(5 \times 10^3\) becomes 5000 in standard notation. Here, 5 is multiplied by 10 three times, moving the decimal point three places to the right.
Converting to standard notation involves:
  • Moving the decimal point to the right if the exponent is positive
  • Moving it to the left if the exponent is negative
This process is important for understanding real-world measurements and quantities better.
Negative Exponent
A negative exponent indicates that you divide by the base number rather than multiply. In the expression \(a \times 10^{-n}\), the negative exponent \(-n\) tells us to move the decimal point \(n\) places to the left.
For instance, in \(-7.68 \times 10^{-4}\), the \-4\ exponent means we need to make the number smaller. To achieve this, we shift the decimal point four places to the left, turning \(-7.68\) into \-0.000768\.
Remembering the role of the negative exponent helps in accurately converting numbers into their standard forms.
Decimal Point
The decimal point plays a crucial role in numerical conversions. It separates the whole number part from the fractional part. In scientific notation, moving the decimal point to the left or right alters the magnitude of the number.
For instance, moving the decimal four places left in \(-7.68\) results in \-0.000768\. Here's how the movement works:
  • Positive exponents move the decimal to the right, making the number larger.
  • Negative exponents move the decimal to the left, making the number smaller.
Properly handling the decimal point ensures accurate conversion between scientific and standard notations.

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Most popular questions from this chapter

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{4}{5}\right)^{-2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{2\left(-m^{-1}\right)^{-4}}{9\left(m^{-3}\right)^{2}} $$

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{1}{4 x}\right)^{-2} $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (g h)(-3) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(3 r^{2}\right)^{2} r^{-5}}{r^{-2} r^{3}}\left(2 r^{-6}\right)^{2} $$
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