Apply the properties of integer exponents to generate an equivalent expression to (7^-3)^7 x 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 need to simplify each expression separately and then combine them.
First, let's simplify (7^-3)^7:
Using the property (a^m)^n = a^(m*n), we can rewrite it as 7^(-3*7) = 7^-21.
Next, let's simplify 7^20:
Since we need to convert it to a positive exponent, we can write it as (7^20/1) = (7^20/1^6) using the property a^0 = 1. Then, applying the exponent property (a^m)^n = a^(m*n), we get 7^(20*1/6) = 7^(20/6) = 7^(10/3).
Now, let's simplify 6^-2:
Using the property a^(-m) = 1/(a^m), we get 6^-2 = 1/(6^2) = 1/36.
Combining the simplified expressions, we have:
(7^-3)^7 x 7^20/6^-2 = 7^-21 x 7^(10/3) x (1/36).
To multiply expressions with the same base, we can add their exponents, so we have:
7^(-21 + 10/3) x (1/36).
To rewrite the exponent as an improper fraction, we have (10/3) = 3 1/3 = 10/3.
Now, we can combine the fractions in the exponent:
7^(10/3) = 7^(3 1/3) = 7^(3 + 1/3) = 7^3 x 7^(1/3) = 343 x (7^(1/3)).
Therefore, the equivalent expression with only positive exponents is:
(7^-3)^7 x 7^20/6^-2 = 343 x (7^(1/3)) x (1/36).
Now, we can evaluate the expression:
343 x (7^(1/3)) x (1/36) = 343 x (7^(1/3))/36 = 343/36 x (7^(1/3)).
So, the answer as an improper fraction is 343/36 x (7^(1/3)).
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents.
First, let's simplify the expression (7^-3)^7:
Using the property (a^m)^n = a^(m * n), we can multiply the exponents:
(7^-3)^7 = 7^(-3 * 7) = 7^-21
Next, let's simplify the expression 7^20:
Now, let's simplify the expression (6^-2):
Using the property (a^-m) = 1 / (a^m), we can write 6^-2 as 1 / 6^2.
Now, let's rewrite the expression with only positive exponents:
(7^-3)^7 x 7^20 / 6^-2 = 7^-21 x 7^20 / (1 / 6^2) = 7^-21 x 7^20 x 6^2
Now, to solve the expression, we can apply the product of powers property.
The product of powers property states that a^m x a^n = a^(m + n).
Using this property, we can add the exponents:
7^-21 x 7^20 x 6^2 = 7^(-21 + 20) x 6^2 = 7^-1 x 6^2
Now, we can further simplify by applying the negative exponent property:
The negative exponent property states that a^-n = 1 / a^n.
So, 7^-1 = 1 / 7^1 = 1 / 7.
Now, the expression becomes:
1 / 7 x 6^2 = 1 / 7 x 36 = 36 / 7
Therefore, the answer as an improper fraction is 36/7.
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents:
1. Power of a Power: When we raise a power to another exponent, we multiply the exponents.
(a^m)^n = a^(m*n)
2. Product of Powers: When we multiply two powers with the same base, we add the exponents.
a^m * a^n = a^(m+n)
3. Quotient of Powers: When we divide two powers with the same base, we subtract the exponents.
a^m / a^n = a^(m-n)
Now let's apply these properties to the given expression, (7^-3)^7 × 7^20 / 6^-2:
Step 1: Apply the Power of a Power property to simplify the exponents in (7^-3)^7.
(7^-3)^7 = 7^(-3*7) = 7^-21
Step 2: Apply the Product of Powers property to simplify the multiplication of the exponents.
7^-21 × 7^20 = 7^(-21+20) = 7^-1
Step 3: Apply the Quotient of Powers property to simplify the division of the exponents.
7^-1 / 6^-2 = 7^(-1-(-2)) = 7^1 = 7
So, the equivalent expression with positive exponents is 7. Now, to convert it to an improper fraction:
7 can be written as 7/1. Therefore, the answer as an improper fraction is 7/1.