What is the fourth root of 4^4^4?
A long wire is cut into three smaller pieces in the ratio 7:3:2. If the shortest piece is 16 cm, what is the area of the largest rectangle that can be created using the longest piece?
Find the value of x in the equation 2013^2013 - 2013^2012= 2012*2013^x.
Thank you!
To find the fourth root of 4^4^4, we need to understand the order of operations in this expression. The exponentiation operation is performed from right to left, so we first need to calculate 4^4^4.
To calculate 4^4^4, we start by solving the innermost exponentiation operation, which is 4^4. This equals 256. Now we replace 4^4 with 256 in the expression 4^4^4.
So, we have 4^256. To find the fourth root of this, we raise it to the power of 1/4.
Using a calculator, or a mathematical software, we can find the fourth root of 4^4^4.
Now moving on to the second question, we need to find the length of the longest piece of wire.
The ratio of the lengths of the three pieces is 7:3:2.
To find the length of the longest piece, we first need to find the total length of the wire.
We can do this by adding the lengths of the three pieces together.
Let's say the lengths of the three pieces are 7x, 3x, and 2x.
Given that the shortest piece is 16 cm, we can set up the following equation:
2x = 16
Solving for x, we find x = 8.
Now, substituting this value of x back into the lengths of the three pieces, we have:
Longest piece = 7x = 7 * 8 = 56 cm
To find the area of the largest rectangle that can be created using the longest piece, we need to consider that a rectangle's area is given by length * width.
Since we only have the length of the longest piece, we need to find its corresponding width.
The maximum area of a rectangle occurs when its length and width are equal, which means the largest rectangle is a square.
So, the width of the largest rectangle is also 56 cm.
Therefore, the area of the largest rectangle is 56 cm * 56 cm = 3136 square cm.
Now let's move on to the third question.
We need to find the value of x in the equation 2013^2013 - 2013^2012 = 2012 * 2013^x.
To solve for x, we need to isolate it on one side of the equation.
First, let's subtract 2013^2012 from both sides:
2013^2013 - 2013^2012 - 2013^2012 = 2012 * 2013^x - 2013^2012
Simplifying:
2013^2013 - 2 * 2013^2012 = 2012 * 2013^x - 2013^2012
Next, let's add 2013^2012 to both sides:
2013^2013 = 2012 * 2013^x
Now, let's divide both sides by 2012 to isolate the term with x:
2013^2013 / 2012 = 2013^x
Finally, we need to take the logarithm base 2013 of both sides to solve for x:
log base 2013 (2013^2013 / 2012) = log base 2013 (2013^x)
Using logarithmic properties, we can simplify this to:
2013 = x
Therefore, the value of x is 2013.
I hope this explanation helps!