Dividing and Multiplying in Standard Form

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Questions and Answers

What is the structure of a number in standard form?

The structure is written as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

In dividing numbers in standard form, what do you do with the exponents?

You subtract the exponent of the divisor from the exponent of the dividend.

Given $6 \times 10^3$ divided by $3 \times 10^1$, what is the result when converted to standard form?

The result is $2 \times 10^2$.

When multiplying in standard form, what is done with the coefficients?

You multiply the coefficients together. Signup and view all the answers

If you multiply $2 \times 10^2$ by $5 \times 10^3$, what is the result before adjustment?

The result is $10 \times 10^{5}$. Signup and view all the answers

What must you do if the coefficient in the multiplication result exceeds 10?

You must adjust it to ensure the coefficient is between 1 and 10. Signup and view all the answers

What is the adjusted form of $10 \times 10^5$?

The adjusted form is $1 \times 10^{6}$. Signup and view all the answers

What steps are involved in dividing numbers written in standard form?

Divide the coefficients, subtract the exponents, and adjust to standard form if necessary. Signup and view all the answers

What is the rule for the coefficients in standard form?

The coefficients must be greater than or equal to 1 and less than 10. Signup and view all the answers

When dividing the numbers in standard form, which operation is performed first?

Divide the coefficients Signup and view all the answers

What is the combined result of dividing (8.5 × 10^4) by (1.7 × 10^2) in standard form?

5.0 × 10^3 Signup and view all the answers

If you multiply (7.2 × 10^1) by (2.5 × 10^2), what is the proper way to express the result in standard form?

1.80 × 10^3 Signup and view all the answers

What should be done after multiplying the coefficients in standard form?

Add the exponents Signup and view all the answers

What is the exponent adjustment needed when you multiply (3.6 × 10^3) by (4.0 × 10^2) and get 14.4?

Add 1 to the exponent Signup and view all the answers

What final result do you obtain when dividing (5.0 × 10^5) by (2.5 × 10^3)?

2.0 × 10^3 Signup and view all the answers

Which step is NOT part of the process when multiplying in standard form?

Divide the exponents Signup and view all the answers

Study Notes

Dividing In Standard Form

Standard Form Structure: Written as ( a \times 10^n ), where:
- ( 1 \leq a < 10 )
- ( n ) is an integer
Steps for Division:
1. Divide the coefficients: Divide ( a_1 ) by ( a_2 ).
2. Subtract the exponents: For the powers of ten, subtract the exponent of the divisor from the exponent of the dividend:
  - If ( a_1 \times 10^{n_1} \div a_2 \times 10^{n_2} ), then:
  - Result = ( \frac{a_1}{a_2} \times 10^{n_1 - n_2} )
3. Adjust to standard form: If the coefficient is not between 1 and 10, adjust it accordingly.
Example:
- Divide ( 6 \times 10^3 ) by ( 3 \times 10^1 ):
  - Coefficient: ( \frac{6}{3} = 2 )
  - Exponents: ( 3 - 1 = 2 )
  - Result: ( 2 \times 10^2 )

Multiplying In Standard Form

Steps for Multiplication:
1. Multiply the coefficients: Multiply ( a_1 ) by ( a_2 ).
2. Add the exponents: For the powers of ten:
  - If ( a_1 \times 10^{n_1} \times a_2 \times 10^{n_2} ), then:
  - Result = ( (a_1 \times a_2) \times 10^{n_1 + n_2} )
3. Adjust to standard form: If the coefficient exceeds 10 or is less than 1, adjust to standard form.
Example:
- Multiply ( 2 \times 10^2 ) by ( 5 \times 10^3 ):
  - Coefficient: ( 2 \times 5 = 10 ) (needs adjustment)
  - Exponents: ( 2 + 3 = 5 )
  - Adjusted Result: ( 1 \times 10^{5} ) (where ( 10 = 1 \times 10^1 ), hence ( 1 \times 10^{6} ))

Standard Form Structure

Standard form is written as ( a \times 10^n ) where ( a ) is between 1 and 10 (inclusive) and ( n ) is an integer.

Dividing Numbers in Standard Form

To divide numbers in standard form, divide the coefficients and subtract the exponents of the powers of ten.
The standard form result can be achieved when ( \frac{a_1}{a_2} ) is adjusted to be between 1 and 10.

Multiplying Numbers in Standard Form

To multiply numbers in standard form, multiply the coefficients and add the exponents of the powers of ten.
The standard form result can be achieved when ( a_1 \times a_2 ) is adjusted to be between 1 and 10.

Standard Form

Standard form expresses numbers as a × 10^n, where 1 ≤ a < 10 and n is an integer.
The coefficient (a) is a number between 1 and 10 (not including 10).
The exponent (n) is an integer.

Dividing in Standard Form

To divide numbers in standard form, follow these steps:
- Divide the coefficients.
- Subtract the exponents.
- Combine the results in standard form.

Multiplying in Standard Form

To multiply numbers in standard form, follow these steps:
- Multiply the coefficients.
- Add the exponents.
- Combine the results in standard form.
- Adjust the coefficient and exponent of the result to ensure the coefficient is between 1 and 10.

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Description

This quiz covers the essential steps for dividing and multiplying numbers in standard form. It includes the structure of standard form, detailed steps for both operations, and examples to illustrate the process. Test your understanding of how to handle coefficients and exponents effectively.