Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)7 ⋅ 7206−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
To rewrite the expression with only positive exponents, we can use the properties of integer exponents:
1) Any number raised to the power of 0 is equal to 1: a^0 = 1
2) When multiplying two numbers with the same base, we add the exponents: a^m * a^n = a^(m+n)
3) When dividing two numbers with the same base, we subtract the exponents: a^m / a^n = a^(m-n)
Let's breakdown the given expression step by step:
Step 1: (7^(-3)) * 7 * (7^(2)) * (20^6 * 2^(-2))
Step 2: (7^(-3)) * 7 * (7^(2)) * (20^6 / 2^2)
Step 3: (1/7^3) * 7 * (7^2) * (20^6 / 2^2)
Step 4: (1/7^3) * 7 * (7^2) * (20^6 / 2^2)
Step 5: (1/7^3) * 7 * (7^2) * (20^6 / 2^2)
Step 6: (1/7^3) * 7 * (7^2) * (20^6 / 2^2)
Step 7: (1/7^3) * 7 * 7^2 * 20^6 / 2^3
Step 8: (1/7^3) * 7^3 * (20^6 / 2^3)
Step 9: (1/7^3) * 7^3 * (10^3)^6 / 2^3
Step 10: (1/7^3) * 7^3 * 10^(3*6) / 2^3
Step 11: (1/7^3) * 7^3 * 10^18 / 2^3
Now, let's simplify the expression:
Step 12: (1/7^3) * 7^3 * 10^18 / 2^3 = 1 * 7^3 * 10^18 / 2^3
Step 13: 7^3 * 10^18 / 2^3 = (7 * 7 * 7 * 10^18) / (2 * 2 * 2)
Step 14: (7 * 7 * 7 * 10^18) / (2 * 2 * 2) = (343 * 10^18) / 8
Step 15: (343 * 10^18) / 8 = 3430^18 / 8
Step 16: 3430^18 / 8 = (3430^18) / (2^3)
Step 17: (3430^18) / (2^3) = 3430^18 / 8 = 813025390625
Therefore, the simplified expression is 813025390625.
To generate an equivalent expression with only positive exponents for (7−3)7 ⋅ 7206−2, we can use the properties of integer exponents.
First, let's apply the rule that states: a^(-n) = 1 / (a^n) to 7206^-2.
7206^-2 = 1 / (7206^2)
Next, we can apply the rule that states: (a/b)^n = (a^n)/(b^n) to (7−3)7.
(7−3)7 = (4)7 = 4^7
Now, we have the expression: 4^7 * (1 / (7206^2))
To solve this expression, we simplify each part individually.
4^7 = 16384
And (7206^2) = 51883236
Now, substituting the values back into the expression, we have:
16384 * (1 / 51883236)
To simplify this expression, we multiply 16384 by 1 and divide by 51883236:
16384 / 51883236
The final answer, expressed as an improper fraction, is 4096/12970809.
To generate equivalent expressions with only positive exponents, we can apply the following rules for integer exponents:
1. **Product Rule:** When two exponential expressions with the same base are multiplied, we can add their exponents. For example, if we have a^x * a^y, it is equal to a^(x+y).
2. **Quotient Rule:** When two exponential expressions with the same base are divided, we can subtract their exponents. For example, if we have a^x / a^y, it is equal to a^(x-y).
3. **Power Rule:** When we have an exponential expression raised to another exponent, we can multiply the exponents. For example, (a^x)^y is equal to a^(x*y).
Now let's apply these rules to the expression (7-3)7 ⋅ 7206-2 step by step:
Step 1: Simplify (7-3) = 4.
Step 2: Apply the Power Rule to (7206) as we need to raise it to -2:
(7206^(-2)) = (7^2 * 10^6)^(-2) = 7^(-2) * 10^(-12)
Step 3: Apply the Product Rule to (7^(-2) * 10^(-12)) and 4:
(4) * (7^(-2) * 10^(-12)) = 4 * 7^(-2) * 10^(-12)
Now, let's solve the expression:
Step 1: Evaluate 4 * 7^(-2) = 4/7^2 = 4/49
Step 2: Simplify the expression 10^(-12).
Since any non-zero number raised to a negative exponent is equal to its reciprocal raised to the positive exponent, we have:
10^(-12) = (1/10)^(12) = 1/10^12 = 1/1000000000000
Step 3: Multiply the results from Step 1 and Step 2:
(4/49) * (1/1000000000000) = 4/(49 * 1000000000000)
Finally, we write the answer as an improper fraction:
Answer: 4/(49 * 1000000000000)