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Chapter 10: Problem 58

For Exercises \(49-64\), convert to standard form. $$6.13 \times 10^{7}$$

Short Answer

Expert verified
61300000.0

Step by step solution

01

- Understand the problem

The exercise requires converting the given number in scientific notation, which is \(6.13 \times 10^{7}\), into standard form.
02

- Identify the components

In the scientific notation \(6.13 \times 10^{7}\), the coefficient is 6.13 and the exponent is 7.
03

- Shift the decimal point

To convert the number to standard form, shift the decimal point in the coefficient (6.13) to the right based on the exponent. Since the exponent is 7, move the decimal point 7 places to the right.
04

- Move the decimal point

Starting from 6.13, moving the decimal 7 places to the right gives us: \(61300000.0\). So, \(6.13 \times 10^{7} = 61300000.0\).

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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 way of expressing very large or very small numbers in a more manageable form. It typically involves a coefficient and an exponent. The coefficient is a number greater than or equal to 1 but less than 10, and it is multiplied by 10 raised to an exponent. This method simplifies calculations and makes it easier to read and write such numbers.
For example, instead of writing 61,300,000, you write it as \(6.13 \times 10^7\). The coefficient here is 6.13 and the exponent is 7.
standard form
Standard form, also known as decimal form, is the regular way of writing numbers without exponents. It鈥檚 the format we usually see in everyday life. For example, instead of \(6.13 \times 10^7\), you would write 61,300,000. This form is useful because it is straightforward and commonly understood by everyone.
To convert a number from scientific notation to standard form, you simply shift the decimal point in the coefficient based on the exponent.
decimal point shift
Shifting the decimal point is a key step in converting scientific notation to standard form. The movement of the decimal point depends on the exponent value in the scientific notation.
For the number \(6.13 \times 10^7\):
  • The decimal point in the coefficient 6.13 needs to be moved 7 places to the right.
  • If the exponent is positive, move the decimal point to the right.
  • If the exponent is negative, move the decimal point to the left.
Starting from 6.13, moving the decimal point 7 places to the right results in 61,300,000.
exponents
An exponent indicates how many times you multiply a number by 10. In scientific notation, it tells you how many places to move the decimal point. For instance, in \(10^7\), the exponent is 7, meaning you move the decimal point 7 places to the right to convert to standard form.
Conversely, if the exponent is negative, such as in \(10^{-3}\), it means you move the decimal point 3 places to the left. This helps in both magnifying large numbers and compressing small numbers into a simpler form.

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Most popular questions from this chapter

Perform the indicated operations. $$\left(1.2 x^{2}+3.5 x-9.6\right)+\left(2.9 x^{2}-6.7 x-3.5\right)$$

Perform the indicated operations. $$\left(-9 m^{4}-7 m^{2}+8\right)-\left(3 m^{4}+9 m^{3}-11\right)-\left(8 m^{3}+m^{2}-3\right)$$

Simplify. Write the answers with positive exponents only. $$z^{-4} z^{7}$$

Multiply the polynomials. $$\begin{array}{r}9 a^{2}+2 a-4 \\\\\times \quad 4 a^{2}+a+3 \\\\\hline\end{array}$$

Simplify. Write the answers with positive exponents only. $$\left(k^{-7}\right)^{-2}$$
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