What is the value of n such that 2^n=4^4 x 8^8 x 16^16?
Well, let's break down this problem. We know that 4 is equal to 2^2, 8 is equal to 2^3, and 16 is equal to 2^4. So, we can rewrite the equation as 2^n = (2^2)^4 x (2^3)^8 x (2^4)^16. Now, using the property of exponents that (a^b)^c is equal to a^(b x c), we can simplify the equation to 2^n = 2^(2 x 4) x 2^(3 x 8) x 2^(4 x 16).
Since the bases on both sides of the equation are equal (they are both 2), we can simply equate the exponents:
n = 2 x 4 + 3 x 8 + 4 x 16
And when you calculate the sum, you get n = 172. So, the value of n that makes the equation true is 172.
To find the value of n that satisfies the equation 2^n = 4^4 x 8^8 x 16^16, we can rewrite the right side of the equation using the properties of exponents.
First, let's simplify each term on the right side:
4^4 = (2^2)^4 = 2^(2*4) = 2^8
8^8 = (2^3)^8 = 2^(3*8) = 2^24
16^16 = (2^4)^16 = 2^(4*16) = 2^64
Now we can substitute these simplified expressions back into the equation:
2^n = 2^8 x 2^24 x 2^64
To solve for n, we can equate the exponents on both sides of the equation:
n = 8 + 24 + 64
Adding the exponents, we get:
n = 96
Therefore, the value of n that satisfies the equation is 96.
To find the value of n, we need to simplify the right side of the equation 2^n = 4^4 x 8^8 x 16^16.
Let's start by expressing all the numbers as powers of 2:
4 = 2^2
8 = 2^3
16 = 2^4
Substituting these values into the equation, we get:
2^n = (2^2)^4 x (2^3)^8 x (2^4)^16
Applying the power of a power rule, we can simplify the equation further:
2^n = 2^(2x4) x 2^(3x8) x 2^(4x16)
Using the product of powers rule, we can combine the exponents:
2^n = 2^8 x 2^24 x 2^64
To combine these exponents, we can use the sum of powers rule:
2^n = 2^(8 + 24 + 64)
2^n = 2^96
Now, we can equate the exponents:
n = 96
Therefore, the value of n that satisfies the equation 2^n = 4^4 x 8^8 x 16^16 is 96.
4^4 x 8^8 x 16^16 are all powers of 2
= (2^2)^4 x (2^3)^8 x (2^4)^16
= 2^8 x 2^24 x 2^64
= 2^96
so what do you think?