Apply the Properties of Integer Exponents to generate equivalent expressions to (7^-3)^7x7^20/6^-2 ⋅ with only positive exponents, then solve the expression.
To simplify the given expression and apply the properties of integer exponents, we start by multiplying the exponents within the parentheses:
(7^-3)^7 = 7^(-21)
Next, we multiply the exponents for 7^(-21) and 7^20:
7^(-21) ⋅ 7^20 = 7^(-21 + 20) = 7^(-1)
Now, let's simplify the other part of the expression:
6^-2 = 1/6^2 = 1/36
Finally, we can put the simplified expression together:
(7^-3)^7 ⋅ 7^20 / 6^-2 = 7^(-1) ⋅ 7^20 / 1/36 = 7^(-1) ⋅ 7^20 ⋅ 36
Using the property of multiplying exponents with the same base, we can add the exponents:
7^(-1) ⋅ 7^20 = 7^(20-1) = 7^19
Now, our expression becomes:
7^19 ⋅ 36
To solve this expression, we need to know the value of 7^19. Since it's a very large number, it's more practical to use a calculator or computer program to find the result.
To apply the properties of integer exponents and generate equivalent expressions with positive exponents, we can use the following rules:
1. Product rule: a^m * a^n = a^(m+n)
2. Quotient rule: a^m / a^n = a^(m-n)
3. Power rule: (a^m)^n = a^(m*n)
4. Negative exponent rule: a^-n = 1/a^n
Now let's simplify the expression step by step:
(7^-3)^7 * 7^20 / 6^-2
First, let's simplify the exponents within the parentheses:
(7^-3)^7 = 7^(-3*7) = 7^-21
Next, let's simplify the division by applying the quotient rule:
7^-21 * 7^20 = 7^(-21+20) = 7^-1
Now, let's simplify the negative exponent:
7^-1 = 1/7^1 = 1/7
Therefore, the simplified expression with positive exponents is 1/7.
To solve the expression, we have already simplified it to 1/7. There is no further solution needed.
To generate equivalent expressions with positive exponents, we can use the properties of exponents.
The first property states that for any non-zero number a, a^(-n) is equal to 1/a^n. Using this property, we can rewrite (7^-3) as 1/(7^3).
The second property states that when we divide two numbers with the same base, we subtract their exponents. Therefore, we can rewrite 7^20/6^-2 as 7^(20-(-2)) = 7^22.
Now, we can substitute these values into the expression:
(7^-3)^7 * 7^20/6^-2
= (1/(7^3))^7 * 7^22
= 1/(7^(3*7)) * 7^22
= 1/7^21 * 7^22
Now, we can apply another exponent property: when we multiply two numbers with the same base, we add their exponents.
Therefore, we have 1/(7^(21-22)) = 1/7^-1.
Using the first property again, we can rewrite 1/7^-1 as 7^1.
Therefore, the final equivalent expression with only positive exponents is 7.
So, (7^-3)^7 * 7^20/6^-2 = 7.