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Chapter 5: Problem 81

Write each number in decimal notation without the use of exponents. $$9.23 \times 10^{5}$$

Short Answer

Expert verified
The number \(9.23 \times 10^{5}\) in decimal notation without exponent is 923000.

Step by step solution

01

Understanding Exponential Notation

In exponential notation, a number like this: \(a \times 10^{n}\), signifies that 'a' needs to be moved 'n' places to the right if 'n' is positive, or to the left if 'n' is negative. Here we have the number \(9.23 \times 10^{5}\), which means we need to move '9.23' five places towards the right.
02

Moving the Decimal Point

Start with the number 9.23. There are 5 movements to the right, so move the decimal point five places to the right. This will create additional zeros.
03

Write the final answer

After moving the decimal point five places to the right in 9.23, we obtain 923000. So, \(9.23 \times 10^{5}\) in decimal notation is 923000.

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Key Concepts

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

Decimal Notation
Decimal notation is the way of representing numbers using the base 10 system. This is the most common method used for writing numbers in our daily lives. In decimal notation, numbers are composed of digits placed at different positions, each with an associated value that is a power of ten.
  • The digit immediately to the left of the decimal point represents the ones place.
  • To the left of the ones place are the tens, hundreds, thousands, etc.
  • The places to the right of the decimal point represent tenths, hundredths, and so on.
Decimal notation does not involve exponents and displays numbers in their full form. For instance, the number 923000 is in decimal notation and clearly shows the value of each digit according to its position.
Multiplying by Powers of Ten
When multiplying a number by powers of ten, we are scaling the number up or down by a factor of ten for each power. Power of ten is expressed in the form of exponentiation like so: \(10^n\).
  • If \(n\) is positive, the decimal point moves to the right.
  • If \(n\) is negative, the decimal point moves to the left.
This technique allows us to simplify multiplication with large or small numbers by mere movement of the decimal, rather than carrying out lengthy multiplication. For example, \(9.23 \times 10^5\) indicates that we need to multiply 9.23 by 100000, effectively moving the decimal point to the right by five places.
Moving Decimal Points
When dealing with numbers involving powers of ten, moving the decimal point is a practical method to convert them into decimal notation. Here's how you do it:
Start with the given number, in this case, 9.23. The exponent on 10 tells you how many places to move the decimal point. In \(9.23 \times 10^5\), the exponent is 5. Move the decimal point five places to the right.
  • After the first move: you have 92.3
  • After the second move: 923
  • Continue moving and each time a position gets empty on the right, add a zero.
  • Finally, after moving five places: 923000
This way, you efficiently convert numbers from exponential to decimal form, reflecting the true scale or size of the number.

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Most popular questions from this chapter

In each exercise, find the product. $$9 x y\left(3 x y^{2}-y+9\right)$$

Perform the indicated operations. $$(y+1)\left(y^{2}-y+1\right)+(y-1)\left(y^{2}+y+1\right)$$

In each exercise, find the product. $$4 x^{3}\left(4 x^{2}-3 x+1\right)$$

Make Sense? In Exercises \(96-99\), determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I divide monomials by dividing coefficients and subtracting exponents.

In Exercises \(100-103,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{12 x^{3}-6 x}{2 x}=6 x^{2}-6 x$$
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