Simplify each of the following exactly:
(8^(-1000) sqr2^10000)/16^500
log_(8) 2
To simplify the expression (8^(-1000) sqr2^10000)/16^500, we can start by simplifying each term separately.
Let's simplify the first term:
8^(-1000) can be written as 1/(8^1000), since a negative exponent indicates the reciprocal. Now, we can rewrite 1 as (2^0)/(2^0) since 1 equals any number divided by itself. Applying the exponent properties, we get:
(2^0)/(2^0 * 8^1000)
Using the product rule and the fact that 8 = 2^3, we can rewrite the denominator as 2^(3 * 1000):
(2^0)/(2^0 * 2^(3 * 1000))
Next, simplify the denominator by adding the exponents of 2:
(2^0)/(2^(0 + 3000))
The exponents in the denominator combine:
2^0/2^3000
Applying the quotient rule, subtract the exponents:
2^(0-3000)
Since any non-zero number raised to the power of 0 is 1, we have:
1/2^3000
Now, let's handle the second term:
sqr2^10000 can be rewritten as 2^(10000/2) by applying the exponent rule that states (a^b)^c = a^(b*c). Simplifying further, we get:
2^5000
Finally, let's simplify the denominator:
16^500 is the same as (2^4)^500. Applying the exponent rule (a^b)^c = a^(b*c), we have:
2^(4*500)
Simplifying the exponent:
2^2000
Now, we can rewrite the expression as:
(1/2^3000) * 2^5000 / 2^2000
To simplify the expression, we combine the terms with the same base by applying the exponent rule a^b * a^c = a^(b+c):
1/2^(3000) * 2^(5000-2000)
This becomes:
1/2^(3000) * 2^(3000)
Now, using the product rule for exponents, we can add the exponents together:
1/2^(3000 + 3000)
Simplifying the exponent:
1/2^6000
Therefore, the expression (8^(-1000) sqr2^10000)/16^500 simplifies exactly to 1/2^6000.
Moving on to the second part of the question:
To calculate log_(8) 2, we need to determine the power to which 8 must be raised to get 2. Rewriting the expression in exponential form, we have:
8^(log_(8) 2) = 2
So, the logarithm base 8 of 2 is equal to 1.