Question
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.
Answers
GPT 3.5
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.
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.