ndex Notation and Powers of 10
10 to the Power 2
The exponent (or index or power) of a number says
how many times to use the number in a multiplication.
102 means 10 × 10 = 100
(It says 10 is used 2 times in the multiplication)
Example: 103 = 10 × 10 × 10 = 1,000
In words: 103 could be called "10 to the third power", "10 to the power 3" or simply "10 cubed"
Example: 104 = 10 × 10 × 10 × 10 = 10,000
In words: 104 could be called "10 to the fourth power", "10 to the power 4" or "10 to the 4"
Answer:
![\frac{x^2}{25} +\frac{y^2}{16} =1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B25%7D%20%2B%5Cfrac%7By%5E2%7D%7B16%7D%20%3D1)
Step-by-step explanation:
For ellipses, the length of the major axis is represents as:
Major axis = ![2a](https://tex.z-dn.net/?f=2a)
where
is called the semi-major axis.
In this case since the major axis is equal to 10 units:
![10=2a](https://tex.z-dn.net/?f=10%3D2a)
solving for the semi-major axis
:
![a=10/2\\a=5](https://tex.z-dn.net/?f=a%3D10%2F2%5C%5Ca%3D5)
and also the minor axis of an ellipse is represented as:
Minor axis = ![2b](https://tex.z-dn.net/?f=2b)
where
is called the semi-minor axis.
Since the minor axis has a length of 8 units:
![8=2b](https://tex.z-dn.net/?f=8%3D2b)
solving for b:
![b=8/2\\b=4](https://tex.z-dn.net/?f=b%3D8%2F2%5C%5Cb%3D4)
Now we can use the equation for an ellipse centered at the origin (0,0):
![\frac{x^2}{a^2} +\frac{y^2}{b^2} =1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7Ba%5E2%7D%20%2B%5Cfrac%7By%5E2%7D%7Bb%5E2%7D%20%3D1)
and substituting the values for
and
:
![\frac{x^2}{5^2} +\frac{y^2}{4^2} =1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B5%5E2%7D%20%2B%5Cfrac%7By%5E2%7D%7B4%5E2%7D%20%3D1)
and finall we simplify the expression to get the equation of the ellipse:
![\frac{x^2}{25} +\frac{y^2}{16} =1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B25%7D%20%2B%5Cfrac%7By%5E2%7D%7B16%7D%20%3D1)
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