Apply the Property of Negative Integer Exponents to generate an expression equivalent to 5−^9/4−^12 with positive exponents only.
To convert the expression from negative exponents to positive exponents, we can use the property of negative integer exponents which states that a^(-n) is equivalent to 1/a^n.
Applying this property to 5^(-9/4), we get:
5^(-9/4) = 1/5^(9/4)
Applying the property to 4^(-12), we get:
4^(-12) = 1/4^12
So, the expression 5^(-9/4) - 4^(-12) written with positive exponents only is:
1/5^(9/4) - 1/4^12
To apply the Property of Negative Integer Exponents and rewrite the expression 5^(-9/4)^(-12) with positive exponents only, we can use the rule which says that a^(-b) is equivalent to 1/a^b.
So, applying this rule to the given expression, we have:
5^(-9/4)^(-12) = 1/5^(9/4)^(12)
Now, we need to simplify the expression further. Since we have the property (a^b)^c = a^(b*c), we can rewrite 9/4 * 12 as 27:
1/5^(9/4)^(12) = 1/5^(27)
Finally, we can convert the negative exponent to a positive exponent by flipping the base, so 1/5^(27) becomes (5^(27))^(-1):
(5^(27))^(-1) = 5^(-27)
Therefore, the expression equivalent to 5^(-9/4)^(-12) with positive exponents only is 5^(-27).
To apply the property of negative integer exponents and convert the given expression with negative exponents to positive exponents, you can use the following steps:
Step 1: Identify the negative exponents in the expression. In this case, we have 5^(-9/4) and 4^(-12).
Step 2: Apply the property of negative exponents: a^(-n) = 1/a^n. This means that to convert a term with a negative exponent to a positive exponent, we can reciprocate the term and change the sign of the exponent.
Step 3: Apply the property to each term in the expression:
For 5^(-9/4):
Reciprocate the term: 1/5^(-9/4)
Change the sign of the exponent from negative to positive: 1/5^(9/4)
For 4^(-12):
Reciprocate the term: 1/4^(-12)
Change the sign of the exponent from negative to positive: 1/4^(12)
Step 4: Simplify the expression further, if possible.
Since we cannot simplify the expression any further, the final expression equivalent to 5^(-9/4) - 4^(-12) with positive exponents only is:
1/5^(9/4) - 1/4^(12)