apply the properties of integer exponents to generate equivalent expressions to (7^-3)^7.7^20/6^-2 with only positive exponents then solve the expression. Provide the answer as an improper fraction.
To apply the properties of integer exponents, we can start by simplifying the exponents separately in each term of the expression.
For the first term, (7^-3)^7, we can use the property (a^m)^n = a^(m*n). Therefore, (7^-3)^7 simplifies to 7^(-3*7) = 7^-21.
For the second term, 7^20, there are no negative exponents to simplify.
For the third term, 6^-2, we use the property a^-n = 1/a^n. So, 6^-2 can be rewritten as 1/6^2.
Now, let's substitute these simplified terms back into the original expression:
(7^-3)^7 * 7^20 / 6^-2
= 7^-21 * 7^20 / (1/6^2)
= 7^-21 * 7^20 * 6^2
To combine the terms, we use the property a^m * a^n = a^(m+n). Therefore, 7^-21 * 7^20 simplifies to 7^(-21 + 20) = 7^-1 = 1/7.
The expression now becomes: (1/7) * 6^2
Since 6^2 = 36, the final expression is:
(1/7) * 36 = 36/7
Therefore, the answer as an improper fraction is 36/7.
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can follow these steps:
Step 1: Simplify the expression (7^-3)^7.
Using the property of exponentiation, (a^m)^n = a^(m*n), we can simplify the expression as follows:
(7^-3)^7 = 7^(-3*7) = 7^(-21).
Step 2: Simplify the expression 7^20/6^-2.
Using the property of quotient of exponents, a^m / a^n = a^(m-n), we can simplify the expression as follows:
7^20/6^-2 = 7^20 * 6^2.
Step 3: Convert the expression to only positive exponents.
To eliminate negative exponents, we can use the property a^(-n) = 1/a^n. Applying this, we get:
7^(-21) = 1/7^21.
Thus, the expression (7^-3)^7.7^20/6^-2 with only positive exponents is equal to 1/7^21 * 7^20 * 6^2.
Step 4: Solve the expression.
To combine the terms, we can use the property a^m * a^n = a^(m+n). Applying this, we can simplify and solve the expression:
1/7^21 * 7^20 * 6^2 = 1/7^(21 - 20) * 7^20 * 6^2
= 1/7^1 * 7^20 * 6^2
= 1/7 * 7^20 * 6^2
= 7^19 * 36.
Hence, the solution to the expression (7^-3)^7.7^20/6^-2, with only positive exponents, is 7^19 * 36 as the improper fraction.
To simplify the expression (7^-3)^7.7^20/6^-2 using the properties of integer exponents and generate equivalent expressions with positive exponents, follow these steps:
Step 1: Simplify the exponents within the parentheses.
(7^-3)^7 = 7^(-3 * 7) = 7^-21
Step 2: Apply the property of exponents when multiplying terms with the same base: a^m * a^n = a^(m + n).
7^-21 * 7^20 = 7^(-21 + 20) = 7^-1
Step 3: Simplify the exponents of 6^-2.
6^-2 = 1/6^2 = 1/36
Step 4: Combine the two expressions using division: a^m / a^n = a^(m - n).
7^-1 / 1/36 = 7^-1 * 36
Since 7^-1 is the reciprocal of 7^1, 7^-1 * 36 = 1/7 * 36 = 36/7
Therefore, the simplified expression is 36/7 as an improper fraction.