3^-17 / 8^-14
Apply the Property of Negative Integer Exponents
When applying the property of negative integer exponents, we can rewrite any negative exponent as the reciprocal of the positive exponent.
So, 3^-17 can be rewritten as 1/3^17, and 8^-14 can be rewritten as 1/8^14.
Therefore, the expression becomes:
(1/3^17) / (1/8^14)
To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction:
(1/3^17) * (8^14/1)
Now, we can simplify:
8^14 = (2^3)^14 = 2^(3*14) = 2^42
So, the expression simplifies to:
(1/3^17) * (2^42/1)
Now, we can multiply the numerators and multiply the denominators:
(1 * 2^42) / (3^17 * 1)
Simplifying further:
2^42 / 3^17
To apply the Property of Negative Integer Exponents, we can rewrite the expression as:
(1/3^17) / (1/8^14)
Now, we can simplify by dividing the fractions:
(1/3^17) * (8^14/1)
= 8^14 / 3^17
To simplify further, we can rewrite 8^14 as (2^3)^14 and 3^17 as (3^2)^8 * 3,
= (2^3)^14 / ((3^2)^8 * 3)
= 2^(3*14) / (3^(2*8) * 3)
= 2^42 / (3^16 * 3)
= 2^42 / 3^17
So, the simplified expression is 2^42 / 3^17.
To apply the property of negative integer exponents, we can rewrite the expression using positive exponents. The property states that when a term with a negative exponent is moved to the denominator, the exponent becomes positive.
Let's rewrite the expression using positive exponents:
3^-17 / 8^-14
By applying the property of negative exponents, the expression becomes:
(1/3^17) / (1/8^14)
Now, we can simplify this expression by multiplying the numerator and denominator by the inverse of the denominator:
(1/3^17) * (8^14/1)
Simplifying further, we can combine the exponents by multiplying:
1 / (3^17 * 8^14)
To simplify the expression even further, we can rewrite 8 as 2^3:
1 / (3^17 * (2^3)^14)
By applying the exponent rule, (a^m)^n = a^(m * n), we can simplify further:
1 / (3^17 * 2^(3*14))
1 / (3^17 * 2^42)
And this is the simplified form of the expression 3^-17 / 8^-14 using the property of negative integer exponents.