Which of the following is an equivalent expression to 13−5 ⋅13−11 with only positive exponents, generated by applying the properties of exponents?(1 point)
To convert the exponents to positive, we can use the property of exponents that states $(a^b)^c = a^{bc}$.
Applying this property to the expression $13^{-5} \cdot 13^{-11}$, we have $(13^5)^{-1} \cdot (13^{11})^{-1}$.
Since raising a number to the power of -1 is the same as taking the reciprocal of the number, we can simplify this expression to $\frac{1}{13^5} \cdot \frac{1}{13^{11}}$.
Using the property of exponents that states $\frac{a^b}{a^c} = a^{b-c}$, we can simplify this expression further to $\frac{1}{13^{5+11}}$.
Finally, combining the exponents gives us $\frac{1}{13^{16}}$.
Therefore, the equivalent expression with only positive exponents is $\boxed{\frac{1}{13^{16}}}$.
To simplify the expression 13−5 ⋅ 13−11 and rewrite it with positive exponents, we can use the property of exponents that states:
a^m ÷ a^n = a^(m-n)
Using this property, we can rewrite the expression as:
13^(−5 − (−11))
Simplifying the exponent, we get:
13^(−5 + 11)
Adding the exponents, we have:
13^6
Therefore, the equivalent expression to 13−5 ⋅ 13−11 with only positive exponents is 13^6.
To simplify the expression 13−5 ⋅13−11 and rewrite it with positive exponents, we can use the properties of exponents.
The properties of exponents state that for any nonzero number a, a^m ⋅ a^n = a^(m+n). This means that if we have a product of two numbers with the same base, we can add their exponents.
Following this rule, we can simplify the expression as follows:
13−5 ⋅13−11
= 13^(-5) ⋅ 13^(-11) (writing the exponents in expanded form)
= (13^(-5 + -11)) (applying the property of exponents by adding the exponents)
= 13^(-16) (simplifying the exponent)
So, the equivalent expression with positive exponents is 13^(-16).