determine the number of digits in the expansion (2^120)(5^125) without using a calculator or computer
Find the coordinates of the two points that trisect the line segment with endpoints A(2,3) and B (8,2)
Oh boy, math questions! Let's see what we can do here.
For the first question, let's break it down. We have (2^120)(5^125). Now, we know that 2^10 is pretty close to 10^3, so we can rewrite the expression as (10^3)^12 times 5^5 times 5^125. Combining the powers, we get 10^36 times 5^130. Now, when we multiply 10^36 and 5^130, we just have to count the number of digits in the product. To do that, we count the number of zeros in 10^36, which is 36, and add that to the exponent of 5^130. So we get 36 + 130, which is 166. That means the number has 166 digits. No calculator needed, just a pencil and some patience!
Now, for the second question, let's put on our math hats again. We have endpoints A(2,3) and B(8,2), and we need to find the two points that trisect the line segment between them. To trisect means to divide into three equal parts.
First, find the x-coordinate of the first trisecting point. It will be two-thirds of the way between 2 and 8. So, (8-2) times 2/3 plus 2 equals 6.
Next, find the y-coordinate of the first trisecting point. It will be two-thirds of the way between 3 and 2. So, (2-3) times 2/3 plus 3 equals 2 and 1/3.
Therefore, the coordinates of the first trisecting point are (6, 2 and 1/3).
Finally, find the coordinates of the second trisecting point. It will be one-third of the way between 2 and 8. So, (8-2) times 1/3 plus 2 equals 3.
Also, find the y-coordinate. It will be one-third of the way between 3 and 2. So, (2-3) times 1/3 plus 3 equals 2 and 2/3.
Therefore, the coordinates of the second trisecting point are (3, 2 and 2/3).
Voila! We did it without a calculator or computer. Math magic at its finest!
To determine the number of digits in the expansion of (2^120)(5^125) without using a calculator or computer, we can use the rule for multiplying powers with the same base. The rule states that when multiplying numbers with the same base, you add the exponents.
In this case, we have (2^120)(5^125). By applying the rule, we can add the exponents:
120 + 125 = 245
Now, our goal is to find the number of digits in 2^245. We can use a logarithmic property to find this. The number of digits in a number can be determined by taking the logarithm of the number, adding 1 to it, and taking the floor value of the result.
Using this property, we can find the number of digits in 2^245:
Digits = floor(log10(2^245) + 1)
Let's calculate this:
Digits = floor(245 * log10(2) + 1)
Since log10(2) is approximately 0.30103, we can substitute it into the equation:
Digits = floor(245 * 0.30103 + 1)
Digits = floor(73.67035 + 1)
Digits = floor(74.67035)
Therefore, the number of digits in the expansion of (2^120)(5^125) is 74.
Moving on to the next question, we are asked to find the coordinates of the two points that trisect the line segment with endpoints A(2,3) and B(8,2).
To find the coordinates of the point that trisects a line segment, we need to find the coordinates of the two points that divide the segment into three equal parts.
First, we need to find the ratios of the line segment that represent the trisection points. Since we are looking for two points, let's call these points P1 and P2.
The ratio for the first trisection point, P1, can be found by dividing the length of the segment AB by 3. Similarly, the ratio for the second trisection point, P2, can be found by multiplying the length of the segment AB by 2/3.
Distance AB = √[(x2 - x1)^2 + (y2 - y1)^2]
Length of AB = √[(8 - 2)^2 + (2 - 3)^2]
Length of AB = √(6^2 + (-1)^2)
Length of AB = √(36 + 1)
Length of AB = √37
Now, let's find the coordinates for P1:
P1(x, y) = (x1 + ((x2 - x1) / 3), y1 + ((y2 - y1) / 3))
P1(x, y) = (2 + ((8 - 2) / 3), 3 + ((2 - 3) / 3))
P1(x, y) = (2 + (6 / 3), 3 + (-1 / 3))
P1(x, y) = (2 + 2, 3 - 1/3)
P1(x, y) = (4, 2 2/3)
So, the coordinates of P1 are (4, 2 2/3).
Now, let's find the coordinates for P2:
P2(x, y) = (x1 + ((2 * (x2 - x1)) / 3), y1 + ((2 * (y2 - y1)) / 3))
P2(x, y) = (2 + ((2 * (8 - 2)) / 3), 3 + ((2 * (2 - 3)) / 3))
P2(x, y) = (2 + ((2 * 6) / 3), 3 + ((2 * -1) / 3))
P2(x, y) = (2 + (12 / 3), 3 + (-2 / 3))
P2(x, y) = (2 + 4, 3 - 2/3)
P2(x, y) = (6, 2 1/3)
So, the coordinates of P2 are (6, 2 1/3).
Therefore, the coordinates of the two points that trisect the line segment with endpoints A(2,3) and B(8,2) are P1(4, 2 2/3) and P2(6, 2 1/3).
To determine the number of digits in the expansion of (2^120)(5^125) without a calculator or a computer, you can use logarithms and properties of exponents:
First, note that (2^120)(5^125) = (2^120)(2^5)(5^125), which can be simplified as 2^125 * 5^125.
Since we are interested in the number of digits, we can focus on the order of magnitude of this product. We know that the number of digits in a positive integer N is given by ⌊log₁₀(N)⌋ + 1, where ⌊⌋ denotes the floor function.
To determine the number of digits in (2^125)(5^125), we can focus on the exponents.
First, let's consider 2^125. We know that log₁₀(2^125) = 125 * log₁₀(2). Since log₁₀(2) is approximately 0.3010, we can calculate 125 * 0.3010 ≈ 37.625. Since the base 10 logarithm of 2 is between 0 and 1, this means that 2^125 is approximately a number with 38 digits.
Next, let's consider 5^125. We know that log₁₀(5^125) = 125 * log₁₀(5). Since log₁₀(5) is approximately 0.6989, we can calculate 125 * 0.6989 ≈ 87.363. Since the base 10 logarithm of 5 is between 0 and 1, this means that 5^125 is approximately a number with 88 digits.
Finally, we multiply the number of digits in 2^125 by the number of digits in 5^125:
38 digits * 88 digits = 3344 digits.
Therefore, (2^120)(5^125) has approximately 3344 digits.
Moving on to the second question:
To find the coordinates of the two points that trisect the line segment AB with endpoints A(2,3) and B(8,2), we can divide the line segment into three equal parts.
First, we need to find the coordinates of the point that divides the line AB into two equal parts. We can use the midpoint formula to find this point.
The midpoint formula states that if we have two distinct points (x₁, y₁) and (x₂, y₂), the coordinates of the midpoint are given by:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
For AB, we have A(2,3) and B(8,2):
Midpoint = ((2 + 8)/2, (3 + 2)/2)
= (10/2, 5/2)
= (5, 2.5)
So, the first point that trisects the line segment AB is (5, 2.5).
Next, to find the coordinates of the second point that divides the line AB into three equal parts, we need to find the point that is one-third of the distance from A to B.
We can use the section formula to calculate this point:
Section Formula: P = ((2x₁ + x₂)/3, (2y₁ + y₂)/3)
Substituting the coordinates of A(2,3) and B(8,2) into the formula:
P = ((2*2 + 8)/3, (2*3 + 2)/3)
= ((4 + 8)/3, (6 + 2)/3)
= (12/3, 8/3)
= (4, 8/3)
So, the second point that trisects the line segment AB is (4, 8/3).
Therefore, the coordinates of the two points that trisect the line segment AB are (5, 2.5) and (4, 8/3).
(2^120)(5^125)
=(2*5)^120 * 5^5
=10^120 * 3125
Therefore there are 124 digits.
Note:
To multiply by 5, you can multiply by 10 and divide by 2.
5^2=25
5^3=125
5^4=625
5^5=3125
Let the two points be H and K, such that AHKB is a straight line.
H=(Xa+(Xb-Xa)/3,Ya+(Yb-Ya)/3)
K=(Xa+2(Xb-Xa)/3,Ya+2(Yb-Ya)/3)