Apply the Properties of Integer Exponents to generate equivalent expressions to (7^-3)^7 times 7^20/ 6-2
To apply the properties of integer exponents, we will simplify each term separately.
First, let's simplify the term (7^-3)^7:
Using the property of exponentiation, we know that (a^m)^n = a^(m*n). Therefore, we have:
(7^-3)^7 = 7^(-3*7) = 7^-21
Next, let's simplify the term 7^20:
When we raise a number to a negative exponent, it is equivalent to taking the reciprocal of the number raised to the positive exponent. So, we have:
7^20 = 1/(7^-20)
Now, let's simplify the term 6^-2:
Using the property of exponentiation, we know that a^-n = 1/a^n. Therefore, we have:
6^-2 = 1/6^2 = 1/36
Now, let's substitute the simplified terms back into the original expression:
(7^-3)^7 * 7^20 / 6^-2
Substituting (7^-3)^7 = 7^-21, 7^20 = 1/(7^-20), and 6^-2 = 1/36:
= 7^-21 * (1/(7^-20)) / (1/36)
When dividing by a fraction, we can multiply by its reciprocal, so:
= 7^-21 * (1/(7^-20)) * (36/1)
Multiplying fractions, we multiply the numerators together and the denominators together:
= (7^-21 * 1 * 36) / (1 * 7^-20)
Using the property of exponentiation, we can rewrite the expression as:
= 36 / (7^(-21) * 7^20)
When we multiply with the same base, we add the exponents:
= 36 / 7^(-21 + 20)
= 36 / 7^-1
Using the property of exponentiation, we know that a^-n is equivalent to 1/a^n. Therefore:
= 36 * 7^1
= 36 * 7
= 252
Therefore, the equivalent expression to (7^-3)^7 * 7^20 / 6^-2 is 252.
Provide it as an Improper fraction
The equivalent expression is 252, which can also be written as the improper fraction 252/1.
Let's break down the expression step-by-step:
Step 1: Simplify the expression (7^-3)^7:
To apply the properties of integer exponents, we can multiply the exponents inside and outside the parentheses:
(7^-3)^7 = 7^(-3 * 7) = 7^(-21)
Step 2: Simplify the expression 7^20:
No further simplification is needed as there are no negative exponents.
Step 3: Simplify the expression 6-2:
6-2 simplifies to 4.
Now, let's combine all the simplified expressions:
(7^-3)^7 * 7^20 / 6-2 = 7^(-21) * 7^20 / 4
Step 4: Apply the rule of multiplying exponential expressions with the same base:
To multiply exponential expressions with the same base, we can add the exponents:
7^(-21) * 7^20 = 7^(-21 + 20) = 7^(-1)
Step 5: Simplify the expression 7^(-1):
A negative exponent indicates the reciprocal of the base, so 7^(-1) is equal to 1/7.
Now, the expression becomes:
(7^-3)^7 * 7^20 / 6-2 = 1/7 * 4
Step 6: Simplify the expression 1/7 * 4:
Multiplying a fraction by a whole number can be done by multiplying the numerator with the whole number, and leaving the denominator unchanged:
1/7 * 4 = (1 * 4) / 7 = 4/7
Therefore, the equivalent expression to (7^-3)^7 * 7^20 / 6-2 is 4/7.
To generate equivalent expressions for the given expression, we can start by simplifying each component individually and then combining them together.
Step 1: Simplify (7^-3)^7
To simplify this expression, we raise 7 to the power of -3 and then multiply the result by 7.
(7^-3)^7 = (1/7^3)^7 = (1/343)^7
Step 2: Simplify 7^20
Simply raise 7 to the power of 20.
7^20
Step 3: Simplify 6-2
Evaluate the expression 6-2.
6-2 = 4
Now we can combine the simplified components to create an equivalent expression.
Final Equivalent Expression:
(1/343)^7 * 7^20 / 4
Note: If you need an actual numerical value for the expression, you can either substitute 7 for 1/343 and calculate it or use a calculator to find the result.