Apply the Properties of Integer Exponents generate equivalent express to (7^-3)^7 * 7^20/6^-2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.
To generate an equivalent expression with only positive exponents, we can use the property $(a^m)^n = a^{mn}$.
Applying this property to $(7^{-3})^7$, we get $7^{-21}$.
Applying the same property to $\left(\frac{7^{20}}{6^{-2}}\right)$, we get $\frac{7^{20}}{6^{-12}}$.
Now, we can simplify the expression by applying the property $\frac{a^m}{a^n} = a^{m-n}$.
Using this property, we get $\frac{7^{20}}{6^{-12}} = 7^{20-(-12)} = 7^{20+12} = 7^{32}$.
Therefore, the simplified expression is $7^{-21} \cdot 7^{32}$.
Now, we can use the property $a^m \cdot a^n = a^{m+n}$ to multiply $7^{-21} \cdot 7^{32}$.
Using this property, we get $7^{-21} \cdot 7^{32} = 7^{-21+32} = 7^{11}$.
Therefore, the expression simplifies to $7^{11}$.
To express this as an improper fraction, we can write $7^{11}$ as $\frac{7^{11}}{1}$.
Therefore, the answer is $\frac{7^{11}}{1}$.
To generate an equivalent expression with only positive exponents, we will apply the properties of integer exponents.
First, let's simplify (7^-3)^7:
(7^-3)^7 = 7^(-3 * 7)
= 7^(-21)
Now let's simplify 7^20/6^-2:
7^20/6^-2 = 7^20 * 6^2 (since dividing by a negative exponent is the same as multiplying by the positive exponent)
= 7^20 * 6^2/1 (since any number divided by 1 does not change its value)
= 7^20 * 6^2 * 1/1 (multiplying by 1 does not change the value)
To simplify the expression, we can combine the terms:
7^(-21) * 7^20 * 6^2 * 1/1
Now, we can combine the bases with the same exponents:
7^(-21+20) * 6^2 * 1/1
= 7^(-1) * 6^2 * 1/1
Finally, let's simplify 7^(-1):
7^(-1) = 1/7^1
= 1/7
Now we have:
1/7 * 6^2 * 1/1
Simplifying further, we get:
1/7 * 36
Multiplying the fractions, we obtain:
36/7
Therefore, the expression (7^-3)^7 * 7^20/6^-2 simplified to 36/7.
To apply the properties of integer exponents and generate an equivalent expression with only positive exponents, we need to follow these steps:
Step 1: Simplify the expression inside the parentheses by multiplying the exponents:
(7^-3)^7 = 7^(-3 * 7) = 7^-21
Step 2: Simplify the fractional part by multiplying the exponents:
6^-2 = 1/6^2 = 1/36
Now, we can rewrite the expression with positive exponents:
7^-3^7 * 7^20/6^-2 = 7^-21 * 7^20 / 1/36
To solve the expression, we can now apply the property of multiplying exponents with the same base:
7^-21 * 7^20 = 7^(-21 + 20) = 7^(-1) = 1/7
Finally, we can rewrite the expression and provide the answer as an improper fraction:
(7^-3)^7 * 7^20 / 6^-2 = (1/7) * (1/36) = 1/7 * 1/36 = 1/252