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.
To generate equivalent expressions with only positive exponents, we need to use the properties of integer exponents.
The expression is 37⋅3−9.
First, we can rewrite 37 as 3^7 since 37 is the same as 3 multiplied by itself 7 times.
Now, using the property a^m⋅a^n = a^(m+n), we can rewrite the expression as 3^7⋅3^-9.
Next, using the property a^m/a^n = a^(m-n), we can simplify the expression further:
3^7⋅3^-9 = 3^(7-9) = 3^-2.
Finally, using the property a^-n = 1/a^n, we can rewrite the expression as 1/3^2 = 1/9.
So, the expression 37⋅3^-9 is equivalent to 1/9.
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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.
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents.
First, we can rewrite 3^(-9) as 1 / 3^9. This is because any number raised to a negative exponent is equivalent to its reciprocal raised to the positive exponent.
Therefore, 37 * 3^(-9) can be expressed as 37 * (1 / 3^9).
Now, let's simplify the expression.
37 * (1 / 3^9)
= 37 / 3^9
To compute 3^9, we can multiply 3 by itself 9 times:
3^9 = 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3
= 19,683
Therefore, 37 / 3^9 simplifies to:
37 / 19,683
This is the final answer, expressed as a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents for the given expression 37⋅3⁻⁹, we need to apply the properties of integer exponents.
The first property states that for any non-zero integer a, a⁰ = 1. Applying this property, we get:
37⋅3⁻⁹ = 37⋅(3⁻⁹)⋅(1)
1 can be expressed as (3⁹) / (3⁹), where 3⁹ is the reciprocal of 3⁻⁹. Now, we can rewrite the expression as:
37⋅(3⁻⁹)⋅(3⁹) / (3⁹)
Using the second property of integer exponents, (a^n)⋅(a^m) = a^(n + m), we can combine the exponents:
37⋅(3⁻⁹)⋅(3³) / (3⁹)
Now, we can use another property of integer exponents: (a^n) / (a^m) = a^(n - m). Applying this property, we simplify the expression further:
37⋅(3⁻⁶)
At this point, we have successfully transformed the given expression into an equivalent expression with only positive exponents. Now, let's solve the simplified expression:
37⋅(3⁻⁶) = 37 / (3⁶)
Calculating 3⁶ = 3 * 3 * 3 * 3 * 3 * 3, we get:
37 / (3 * 3 * 3 * 3 * 3 * 3)
This can be further simplified by canceling out common factors:
37 / 729
So, the solution to the expression 37⋅3⁻⁹, with only positive exponents, is 37/729 as a simplified fraction with no remaining exponents.