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.(1 point)
(7 ^-3) ^ 7 * 7 ^ 20 / 6 ^-2 = ?
To apply the properties of integer exponents, we can use the following rules:
1. When raising a power to another power, multiply the exponents.
2. When dividing two powers with the same base, subtract the exponents.
3. A negative exponent can be converted to a positive exponent by taking the reciprocal of the base.
Using these rules, we can simplify the expression:
(7^-3)^7 * 7^20 / 6^-2 =
Since (7^-3)^7 = 7^(-3*7) = 7^(-21), and 6^-2 = 1/6^2, we can substitute these values:
7^(-21) * 7^20 / (1/6^2) =
Now, let's simplify further using the rule of multiplying and dividing with the same base:
= 7^(-21 + 20) / (1/6^2)
= 7^(-1) / (1/6^2)
= 1/7^1 * 6^2
= 6^2/7
= 36/7
Therefore, the simplified expression is 36/7.
To apply the properties of integer exponents to generate equivalent expressions with only positive exponents, we can use the following rules:
1. Power of a Power: (x^m)^n = x^(m*n)
2. Product of Powers: (x^m) * (x^n) = x^(m+n)
3. Quotient of Powers: (x^m) / (x^n) = x^(m-n)
4. Negative Exponent Rule: x^-n = 1 / x^n
Now let's apply these rules step by step to simplify the expression (7^-3)^7 * 7^20 / 6^-2:
Step 1: Apply the Power of a Power rule:
(7^-3)^7 = 7^(-3*7) = 7^-21
Step 2: Apply the Negative Exponent Rule:
7^-21 = 1 / 7^21
Step 3: Combine the exponents using the Product of Powers rule:
1 / 7^21 * 7^20 = 7^(20-21) = 7^-1
Step 4: Apply the Negative Exponent Rule:
7^-1 = 1 / 7^1 = 1/7
Therefore, the expression (7^-3)^7 * 7^20 / 6^-2 is equivalent to 1/7.
Hence, the answer to the expression is 1/7, which is the improper fraction representation.
To generate equivalent expressions with positive exponents, we need to apply the properties of integer exponents.
1. First, let's simplify the expression (7 ^ -3) ^ 7:
Remember that when we raise a power to another power, we multiply the exponents.
(7 ^ -3) ^ 7 = 7 ^ (-3 * 7) = 7 ^ -21
2. Now, let's simplify the expression 7 ^ 20:
Multiplying exponents with the same base means adding the exponents.
7 ^ 20 = 7 ^ (7 + 13) = 7 ^ 7 * 7 ^ 13
3. Lastly, let's simplify the expression 6 ^ -2:
A negative exponent indicates that we take the reciprocal of the base raised to the positive exponent.
6 ^ -2 = 1 / 6 ^ 2
Now, let's substitute the simplified expressions back into our original expression:
(7 ^ -3) ^ 7 * 7 ^ 20 / 6 ^ -2
= (7 ^ -21) * (7 ^ 7 * 7 ^ 13) / (1 / 6 ^ 2)
= 7 ^ (-21) * 7 ^ 7 * 7 ^ 13 * 6 ^ 2
= 7 ^ (-21 + 7 + 13) * 6 ^ 2
= 7 ^ (-1) * 6 ^ 2
= 1 / 7 * 36
= 36 / 7
Therefore, the answer to the expression (7 ^ -3) ^ 7 * 7 ^ 20 / 6 ^ -2, using positive exponents and written as an improper fraction, is 36/7.