apply the property of negative integer exponents to generate an expression equivalate to 3-27/8-14
To apply the property of negative integer exponents, we need to flip the base and change the sign of the exponent.
The given expression is 3^(-27) / 8^(-14).
To apply the property, we can rewrite it as (1/3)^(27) / (1/8)^(14).
Now, let's simplify these fractions:
(1/3)^(27) = 1^27 / 3^27 = 1 / 3^27
(1/8)^(14) = 1^14 / 8^14 = 1 / 8^14
Putting these simplified fractions back into the expression, we have:
1 / 3^27 / 1 / 8^14
Now, division of fractions can be simplified by multiplying the numerator of the first fraction with the denominator of the second fraction, and the denominator of the first fraction with the numerator of the second fraction:
(1 / 3^27) * (8^14 / 1)
Simplifying this further:
8^14 / 3^27
To apply the property of negative integer exponents, we need to rewrite the given expression in a way that has only positive exponents. Let's break it down step-by-step:
Step 1: Simplify the expression.
To simplify the given expression 3-27/8-14, we can simplify the numerator and denominator separately.
Numerator: 3-27
Here, we have a negative exponent. Using the property of negative exponents, we can rewrite it as 1/3^27.
Denominator: 8-14
Similarly, we have another negative exponent. Using the same property, we can rewrite it as 1/8^14.
Now, let's combine the numerator and denominator:
1/3^27 divided by 1/8^14
Step 2: Apply the rule for dividing exponents.
When dividing two expressions with the same base, we can subtract the exponents.
In this case, we have 1/3^27 divided by 1/8^14.
Using the rule for dividing exponents, we can rewrite it as:
1/3^27 * 8^14
Step 3: Simplify the expression further.
To simplify the expression, we calculate the powers:
1/3^27 = 1/19683 (since 3^27 = 19683)
8^14 = 8192 (since 8^14 = 8192)
So, the final simplified expression equivalent to 3-27/8-14 is 1/19683 * 8192.
To apply the property of negative integer exponents, we need to rewrite the given expression using positive exponents. Let's break it down step by step:
The given expression is: 3^(-27) / 8^(-14)
To make the exponents positive, we can use the property that states: a^(-n) = 1 / a^n.
So, let's rewrite the expression using this property:
(1 / 3^27) / (1 / 8^14)
Now, we can simplify further by multiplying the numerator and the denominator by the reciprocal of the denominator, which is equivalent to multiplying by 8^14:
(1 / 3^27) * (8^14 / 1)
Now, we can simplify the exponents:
1 / 3^27 = 1 / (3 * 3 * 3 * ... * 3) = 1 / 3^(27)
8^14 = (2^3)^14 = 2^(3 * 14) = 2^42
Substituting the simplified exponents:
(1 / 3^27) * (8^14 / 1) = (1 / 3^27) * (2^42)
Now, the expression is equivalent to 1 / 3^27 * 2^42.