Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3^−17/8^−14 with positive exponents only.
To apply the property of negative integer exponents, we have to reciprocate the base and change the sign of the exponent.
Therefore, 3^(-17) can be rewritten as 1/3^(17) and 8^(-14) can be rewritten as 1/8^(14).
Using these equivalences, the expression 3^(-17)/8^(-14) becomes:
(1/3^17)/(1/8^14)
When we divide by a fraction, we multiply by its reciprocal. So, we can rewrite the division as multiplication by the reciprocal:
(1/3^17) * (8^14/1)
Multiplying the numerators and denominators gives us:
(1 * 8^14) / (3^17 * 1)
Simplifying further, we have:
8^14 / 3^17
And that is the expression with positive exponents only.
To apply the property of negative integer exponents, we need to rewrite the given expression with positive exponents only.
The property states:
For any nonzero number a and any positive integer n,
a^(-n) = 1 / a^n
Using this property, let's rewrite the given expression 3^(-17) / 8^(-14):
3^(-17) / 8^(-14) = (1 / 3^17) / (1 / 8^14)
Now, we can simplify further:
(1 / 3^17) / (1 / 8^14) = (8^14 / 3^17)
Therefore, the expression equivalent to 3^(-17) / 8^(-14) with positive exponents only is 8^14 / 3^17.
To apply the property of negative integer exponents, we can rewrite the expression with positive exponents by taking the reciprocal of each base and changing the sign of the exponent.
Given: 3^(-17) / 8^(-14)
Step 1: Apply the property to the numerator:
3^(-17) = 1 / 3^17
Step 2: Apply the property to the denominator:
8^(-14) = 1 / 8^14
Step 3: Substitute the results from step 1 and step 2 back into the original expression:
(1 / 3^17) / (1 / 8^14)
Step 4: Simplify the expression:
Since dividing by a fraction is the same as multiplying by the reciprocal, we can rewrite the expression as:
(1 / 3^17) * (8^14 / 1)
Step 5: Combine the terms:
1 * 8^14 / 3^17
Final expression with positive exponents: 8^14 / 3^17