Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3)7 ⋅ 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 and generate equivalent expressions with only positive exponents, we'll use the following rules:
1. Negative exponent rule: Any expression with a negative exponent can be rewritten by taking the reciprocal of the base and changing the sign of the exponent. For example, x^-3 = 1/x^3.
2. Product of powers rule: When multiplying terms with the same base, we can add their exponents. For example, x^2 * x^3 = x^(2+3) = x^5.
3. Quotient of powers rule: When dividing terms with the same base, we can subtract their exponents. For example, x^5 / x^3 = x^(5-3) = x^2.
Let's apply these rules to (7^-3)7 ⋅ 7^20/6^-2:
First, let's apply the negative exponent rule to (7^-3), which gives us 1/7^3.
Next, let's multiply this expression by 7. Using the product of powers rule, we have:
(1/7^3) * 7 = 7^(1-3) = 7^-2.
Now, let's simplify 6^-2. Using the negative exponent rule, we have 1/6^2.
Next, let's divide 7^20 by 6^2. Using the quotient of powers rule, we have:
7^20 / 6^2 = 7^(20-2) = 7^18.
Putting it all together, our expression becomes:
(7^-3)7 ⋅ 7^20/6^−2 = (1/7^3) * 7 * 7^20 / (1/6^2) = 7^-2 * 7^20 / (1/6^2) = 7^-2 * 7^20 * 6^2.
Now, let's simplify further by adding the exponents:
7^-2 * 7^20 * 6^2 = 7^(-2+20) * 6^2 = 7^18 * 6^2.
To solve this expression, we need to evaluate the values of 7^18 and 6^2.
7^18 = 2,824,752,475
6^2 = 36
Therefore, the answer is:
7^(-2+20) * 6^2 = 2,824,752,475 * 36 = 101,689,692,700.
So the expression simplifies to 101,689,692,700.
As an improper fraction, the answer is 101689692700/1.
To generate equivalent expressions with only positive exponents, let's apply the properties of integer exponents step-by-step:
1. First, we apply the property that a negative exponent is equivalent to the reciprocal of the base with a positive exponent. So, (7^−3) becomes 1/(7^3).
2. Next, we use the property of multiplying exponents when the bases are the same. So, (7^−3)7 becomes (1/(7^3)) * 7 = (7^1)/(7^3).
3. Applying the property of dividing exponents when the bases are the same, we subtract the exponents in the numerator and denominator. So, (7^1)/(7^3) becomes 7^(1-3) = 7^(-2).
4. Now, we can rewrite 6^(-2) as 1/(6^2) using the property of negative exponents.
Putting it all together, the expression (7^−3)7 ⋅ 7^20/6^−2 becomes (7^(-2)) * (7^20) / (1/(6^2)).
Simplifying further, we have (7^(-2)) * (7^20) * (6^2).
Using the property of multiplying exponents with the same base, we add the exponents: (7^(-2)) * (7^20) * (6^2) = 7^(-2+20) * 6^2 = 7^18 * 6^2.
Now, we evaluate 7^18 and 6^2:
7^18 = 2,824,752,200,000,000,000
6^2 = 36
Therefore, the solution to the expression is 2,824,752,200,000,000,000/36, which simplifies to 78,465,339,877,777,777.
To apply the properties of integer exponents and generate equivalent expressions with positive exponents, let's break down the given expression step by step:
1. Start with the expression (7^−3)7 ⋅ 7^20/6^−2.
2. Apply the rule that states: a^(-n) = 1 / (a^n). The negative exponent indicates taking the reciprocal of the base raised to the positive exponent. We can rewrite 7^−3 as 1 / 7^3.
3. We now have (1 / 7^3)7 ⋅ 7^20/6^−2.
4. Multiply the bases with the same exponent by adding the exponents. We can rewrite (1 / 7^3)7 as 1 / 7^(3-1) = 1 / 7^2.
5. Our expression becomes (1 / 7^2) ⋅ 7^20/6^−2.
6. Apply the rule that states: a^m / a^n = a^(m-n). We can rewrite 7^20/6^−2 as 7^(20-(-2)) = 7^22.
7. Now, our expression is (1 / 7^2) ⋅ 7^22.
8. Apply the product of powers rule, which states: a^m ⋅ a^n = a^(m+n). We can rewrite (1 / 7^2) ⋅ 7^22 as 1 / 7^2 ⋅ 7^22 = 1 / 7^(2-(-22)) = 1 / 7^24.
9. Our final expression is now 1 / 7^24.
To solve this expression, we rewrite it as an improper fraction:
1 / 7^24 = 1 / (7^24/1) = 1 ÷ (1/7^24) = 1 × (7^24/1) = 7^24/1 = 7^24.
Therefore, the solution to the expression (7^−3)7 ⋅ 7^20/6^−2 as an improper fraction is 7^24.