Which of the following is an equivalent expression to 13−5 ⋅13−11 with only positive exponents, generated by applying the properties of exponents?
The expression 13−5 can be rewritten as 1/13^5, and 13−11 can be rewritten as 1/13^11.
Therefore, 13−5 ⋅13−11 can be rewritten as (1/13^5) ⋅ (1/13^11).
Applying the property that states a^m ⋅ a^n = a^(m+n), we can combine the exponents:
(1/13^5) ⋅ (1/13^11) = 1/13^(5+11) = 1/13^16
So, an equivalent expression would be 1/13^16.
To simplify the expression 13−5 ⋅ 13−11, we can use the property of exponents that states x−n is equal to 1/xn.
Step 1: Rewrite the expression using positive exponents:
13−5 ⋅ 13−11 = (1/135) ⋅ (1/1311)
Step 2: Simplify the expression:
(1/135) ⋅ (1/1311) = 1/(135 ⋅ 1311)
Therefore, the equivalent expression with only positive exponents is 1/(135 ⋅ 1311).
To simplify the expression 13−5 ⋅13−11 and rewrite it using only positive exponents, we need to apply the properties of exponents. The properties we can use are:
1. Product of Powers: a^m ⋅ a^n = a^(m+n)
2. Quotient of Powers: a^m / a^n = a^(m-n)
3. Power of a Power: (a^m)^n = a^(m*n)
4. Power of a Product: (ab)^n = a^n ⋅ b^n
5. Negative Exponent Property: a^(-n) = 1 / a^n
Now let's simplify the given expression step by step:
13−5 ⋅13−11
First, let's rewrite the exponents with positive exponents using the negative exponent property:
13^(-5) ⋅ 13^(-11)
Next, let's combine the bases using the product of powers property:
(13^(-5 - 11))
Simplifying further:
(13^(-16))
Now, if we want to rewrite this with only positive exponents, we can take the reciprocal of the base using the negative exponent property:
1 / (13^16)
So, the equivalent expression with only positive exponents is 1/13^16.
Therefore, the answer is 1/13^16.