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 apply the properties of integer exponents to generate equivalent expressions, we can use the property that for any positive integer a, a^(-n) = 1/a^n.
So, for the expression 3^7 • 3^(-9), we can rewrite it as 3^7 • 1/3^9.
Now, using the property of exponentiation that a^m • a^n = a^(m+n), we can simplify the expression further:
3^7 • 1/3^9 = 3^(7+(-9)) = 3^(-2) = 1/3^2 = 1/9.
Therefore, the simplified fraction is 1/9.
To simplify 3⁷ • 3^‐⁹ and express it with only positive exponents, you can use the property of integer exponents that states: a^m • a^n = a^(m+n).
First, let's simplify the expression:
3⁷ • 3^‐⁹
Using the property of integer exponents, we can combine the exponents:
3⁷ • 3^‐⁹ = 3^(7 + (-9))
Since we're adding a positive exponent to a negative exponent, we can subtract the exponents instead:
3^(7 + (-9)) = 3^(-2)
Now, we have an expression with a negative exponent. To express it with a positive exponent, we can use the property of integer exponents that states: a^(-n) = 1 / a^n.
Applying this property to 3^(-2), we can write it as:
3^(-2) = 1 / (3^2)
Simplifying further, 3^2 = 9. Therefore,
1 / (3^2) = 1 / 9
So, the simplified expression of 3⁷ • 3^‐⁹ with only positive exponents is 1 / 9.
To apply the properties of integer exponents, we can use the rule that states when multiplying with the same base, you add the exponents. In this case, we have 3 raised to the power of 7 multiplied by 3 raised to the power of -9.
First, let's express 3^(-9) as a positive exponent. To do this, we can use the property that states a^-n is equal to 1/a^n. Therefore, 3^(-9) = 1 / (3^9).
Now, let's rewrite the expression using the rule of adding exponents: 3^7 • (1 / 3^9).
Next, let's simplify this expression further. When we multiply, we subtract the exponents. Therefore, 3^7 • (1 / 3^9) = 3^(7 - 9) = 3^(-2).
Finally, let's express 3^(-2) as a positive exponent. Using the rule mentioned earlier, 3^(-2) = 1 / (3^2) = 1 / 9.
So, the equivalent expression of 3^7 • 3^(-9) with only positive exponents is 1/9.
Therefore, the solution to the expression 3^7 • 3^(-9) is the simplified fraction 1/9.