Apply the Properties of Integer Exponents to generate equivalent expressions to 37⋅3−9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
3^7⋅3^−9=
To apply the properties of integer exponents, we can use the rule that states that when multiplying powers with the same base, we add the exponents.
So, 37⋅3^(-9) can be rewritten as (3^7)⋅(3^(-9)).
Using the rule mentioned earlier, we can add the exponents:
3^7⋅3^(-9) = 3^(7 + (-9))
Now, simplifying the exponent:
3^(7 + (-9)) = 3^(-2)
Therefore, 37⋅3^(-9) is equivalent to 3^(-2).
We can further simplify 3^(-2) by using another exponent property, which states that any non-zero number raised to the power of -n is equal to the reciprocal of that number raised to the power of n.
So, 3^(-2) is equivalent to 1/(3^2).
Since 3^2 equals 9, the simplified expression is:
1/9.
apply the properties of integer exponents to generate equivalent expressions to 2^2/2^−4 with only positive exponents, then solve the expression. provide the solution as a whole number.(1 point)
2^2/2^−4=
To generate equivalent expressions with only positive exponents, we can use the rule that states that when dividing powers with the same base, we subtract the exponents.
So, 2^2/2^(-4) can be rewritten as (2^2)/(2^(-4)).
Using the rule mentioned earlier, we can subtract the exponents:
(2^2)/(2^(-4)) = 2^(2 - (-4))
Simplifying the exponent:
2^(2 - (-4)) = 2^(2 + 4)
2^(2 + 4) = 2^6
Therefore, 2^2/2^(-4) is equivalent to 2^6.
Simplifying 2^6, we get:
2^6 = 64
Hence, the solution to the expression 2^2/2^(-4) is 64.
To generate an equivalent expression with only positive exponents, you can use the property of integer exponents which states that for any positive integers a and b,
a^b ⋅ a^c = a^(b+c).
Using this property, we can rewrite the expression as:
37 ⋅ 3^(-9) = 3^7 ⋅ 3^(-9) = 3^(7+(-9)).
Since (-9) is negative, when we rewrite it, it becomes 1/9. Therefore,
3^7 ⋅ 3^(-9) = 3^(7+(-9)) = 3^(-2) or 1/3^2.
Simplifying further, we have:
3^(-2) = 1/3^2 = 1/9.
So, the simplified answer is 1/9.