apply the properties of integer exponents to generate equivalent expression to (7 to the power of -3) to the power of 7 times 7 to the power of 20 divided by 6 to the power of -2
To apply the properties of integer exponents, let's break down the given expression step by step:
1. Start with (7^-3)^7:
According to the property (a^m)^n = a^(m*n), we can multiply the exponents:
(7^-3)^7 = 7^(-3*7) = 7^-21.
2. Then, multiply by 7^20:
According to the property a^m * a^n = a^(m+n), we can add the exponents:
7^-21 * 7^20 = 7^(-21 + 20) = 7^-1.
3. Lastly, divide by 6^-2:
According to the property a^m / a^n = a^(m-n), we can subtract the exponents:
7^-1 / 6^-2 = 7^-1 * 6^2 = 7^-1 * 36.
Therefore, the equivalent expression would be 7^-1 * 36.
To generate an equivalent expression using the properties of integer exponents, we can simplify each part separately and then combine them.
Step 1: Simplify (7^-3)^7
The property of raising a power to a power states that (a^m)^n = a^(m*n). Applying this property, we can simplify (7^-3)^7 as follows:
(7^-3)^7 = 7^(-3*7) = 7^(-21)
Step 2: Simplify 7^20
No further simplification can be done for 7^20.
Step 3: Simplify 6^-2
Using the property of negative exponents, we can rewrite 6^-2 as 1/6^2.
Step 4: Combine the simplified expressions
Now, we can combine the simplified expressions to get the equivalent expression:
(7^-3)^7 * 7^20 / 6^-2
= 7^(-21) * 7^20 / (1/6^2)
= 7^(-21) * 7^20 * 6^2
Using the property of multiplying powers with the same base, we can add the exponents:
= 7^(-21 + 20) * 6^2
= 7^(-1) * 6^2
= (1/7) * 6^2
= 6^2 / 7
Therefore, the equivalent expression is 6^2 / 7.
To generate an equivalent expression using the properties of integer exponents, we will break down the given expression step by step and apply the properties of exponents:
Step 1: Simplify (7^(-3))^7
Using the property of exponentiation of a power, we can multiply the exponents:
(7^(-3))^7 = 7^(-3*7) = 7^(-21)
Step 2: Simplify 7^20
No further simplification is required for this term.
Step 3: Simplify 6^(-2)
Using the property of negative exponents, we can rewrite 6^(-2) as its reciprocal:
6^(-2) = 1/(6^2) = 1/36
Step 4: Putting it all together
Now that we have simplified all the terms, we can combine them using the properties of exponents:
(7^(-3))^7 * 7^20 / 6^(-2) = 7^(-21) * 7^20 * 1/36
To simplify this further, we can use the property of multiplying exponents with the same base:
7^(-21) * 7^20 = 7^(-21 + 20) = 7^(-1) = 1/7
Therefore, the equivalent expression is:
1/7 * 1/36
To multiply fractions, we multiply the numerators and the denominators:
(1 * 1) / (7 * 36) = 1/252
Therefore, the final equivalent expression is 1/252.