Explain patterns in the placement of the decimal point when multiplying 5 by 10^2, 10^1, and 10^0 and when dividing 5 by 10^1, 10^2, and 10^3.
state the products of 5*10^2, 5*10^1, and 5*10^0
state the quotients of 5/10^1, 5/10^2, 5/10^3
When multiplying a number by powers of 10, the placement of the decimal point moves to the right. This is because multiplying by 10 means adding a zero to the end of the number, and each additional power of 10 adds an additional zero.
Let's go through each example:
1. Multiplying 5 by 10^2:
The product of 5 * 10^2 is 500. To get this result, we move the decimal point two places to the right. Since there are no digits to the right of the decimal point in 5, we add two zeros to get 500.
2. Multiplying 5 by 10^1:
The product of 5 * 10^1 is 50. We move the decimal point one place to the right. Since there are no digits to the right of the decimal point in 5, we add one zero to get 50.
3. Multiplying 5 by 10^0:
The product of 5 * 10^0 is 5. Here, 10^0 simply means multiplying by 1, and any number multiplied by 1 remains the same.
Now, let's look at dividing 5 by powers of 10. In division, the decimal point moves to the left because we are essentially removing zeros from the number.
1. Dividing 5 by 10^1:
The quotient of 5 / 10^1 is 0.5. We move the decimal point one place to the left. Since there are no digits to the left of the decimal point in 5, we add a leading zero to get 0.5.
2. Dividing 5 by 10^2:
The quotient of 5 / 10^2 is 0.05. We move the decimal point two places to the left. Since there are no digits to the left of the decimal point in 5, we add two leading zeros to get 0.05.
3. Dividing 5 by 10^3:
The quotient of 5 / 10^3 is 0.005. We move the decimal point three places to the left. Since there are no digits to the left of the decimal point in 5, we add three leading zeros to get 0.005.
So, to summarize:
- When multiplying by 10^n, the decimal point is moved n places to the right, adding zeros if necessary.
- When dividing by 10^n, the decimal point is moved n places to the left, adding leading zeros if necessary.
When multiplying a number by 10 raised to different powers, the decimal point shifts to the right. Here are the patterns:
Multiplying 5 by 10^2:
5 * 10^2 = 5 * 100 = 500
The decimal point shifted two places to the right, resulting in the product 500.
Multiplying 5 by 10^1:
5 * 10^1 = 5 * 10 = 50
The decimal point shifted one place to the right, resulting in the product 50.
Multiplying 5 by 10^0:
5 * 10^0 = 5 * 1 = 5
There is no shift in the decimal point, so the product is 5.
When dividing a number by 10 raised to different powers, the decimal point shifts to the left. Here are the patterns:
Dividing 5 by 10^1:
5 / 10^1 = 5 / 10 = 0.5
The decimal point shifted one place to the left, resulting in the quotient 0.5.
Dividing 5 by 10^2:
5 / 10^2 = 5 / 100 = 0.05
The decimal point shifted two places to the left, resulting in the quotient 0.05.
Dividing 5 by 10^3:
5 / 10^3 = 5 / 1000 = 0.005
The decimal point shifted three places to the left, resulting in the quotient 0.005.