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 simplify the expression and convert all the exponents to positive, we can use the properties of integer exponents:
1) (a^m)^n = a^(m*n)
2) a^(-m) = 1/a^m
3) a^(m/n) = √(a^m)^n if m is divisible by n
Let's apply these properties step by step:
(7^(-3))^7 ⋅ 7^20/6^(-2)
By applying property 1):
(7^(-21)) ⋅ 7^20/6^(-2)
By applying property 2):
(1/7^21) ⋅ 7^20/6^(-2)
By applying property 3):
(1/7^21) ⋅ (sqrt(7^20)) / (sqrt(6^-2))
By simplifying the square roots:
(1/7^21) ⋅ ((7^20)^(1/2)) / ((6^-2)^(1/2))
By applying property 1) again:
(1/7^21) ⋅ 7^(20/2) / 6^(-2/2)
By simplifying exponents:
(1/7^21) ⋅ 7^10 / 6^(-1)
By applying property 2):
(1/7^21) ⋅ 7^10 / (1/6^1)
By simplifying reciprocal:
(1/7^21) ⋅ 7^10 ⋅ 6
By simplifying multiplication:
6 / 7^11
Therefore, the simplified expression is 6 / 7^11, where 6 is the numerator and 7^11 is the denominator.
1) (a^m)^n = a^(m*n)
2) a^(-m) = 1/a^m
3) a^(m/n) = √(a^m)^n if m is divisible by n
Let's apply these properties step by step:
(7^(-3))^7 ⋅ 7^20/6^(-2)
By applying property 1):
(7^(-21)) ⋅ 7^20/6^(-2)
By applying property 2):
(1/7^21) ⋅ 7^20/6^(-2)
By applying property 3):
(1/7^21) ⋅ (sqrt(7^20)) / (sqrt(6^-2))
By simplifying the square roots:
(1/7^21) ⋅ ((7^20)^(1/2)) / ((6^-2)^(1/2))
By applying property 1) again:
(1/7^21) ⋅ 7^(20/2) / 6^(-2/2)
By simplifying exponents:
(1/7^21) ⋅ 7^10 / 6^(-1)
By applying property 2):
(1/7^21) ⋅ 7^10 / (1/6^1)
By simplifying reciprocal:
(1/7^21) ⋅ 7^10 ⋅ 6
By simplifying multiplication:
6 / 7^11
Therefore, the simplified expression is 6 / 7^11, where 6 is the numerator and 7^11 is the denominator.
you sure??
GPT 3.5
I apologize for the mistake in my previous answer. Let's correct it.
We have the expression (7^(-3))^7 ⋅ 7^20/6^(-2).
Using the properties of exponents, we can simplify this expression as follows:
Step 1: Apply the power of a power property (a^m)^n = a^(m*n).
(7^(-3))^7 = 7^((-3)*7) = 7^(-21).
So the expression becomes 7^(-21) ⋅ 7^20/6^(-2).
Step 2: Apply the product of powers property a^m ⋅ a^n = a^(m+n).
7^(-21) ⋅ 7^20 = 7^(-21+20) = 7^(-1).
The expression simplifies to 7^(-1) / 6^(-2).
Step 3: Apply the negative exponent property a^(-m) = 1/a^m.
7^(-1) = 1/7^1 = 1/7.
The expression becomes 1/7 / 6^(-2).
Step 4: Apply the division of powers property a^m / a^n = a^(m-n).
1/7 / 6^(-2) = 1/7 / 6^2 = 1/7 / 36.
Now, divide 1 by 7 and express the result as an improper fraction:
1 ÷ 7 = 1/7.
Therefore, the simplified expression is 1/7 / 36, and the answer as an improper fraction is (1/7) / 36 = 1/252.
We have the expression (7^(-3))^7 ⋅ 7^20/6^(-2).
Using the properties of exponents, we can simplify this expression as follows:
Step 1: Apply the power of a power property (a^m)^n = a^(m*n).
(7^(-3))^7 = 7^((-3)*7) = 7^(-21).
So the expression becomes 7^(-21) ⋅ 7^20/6^(-2).
Step 2: Apply the product of powers property a^m ⋅ a^n = a^(m+n).
7^(-21) ⋅ 7^20 = 7^(-21+20) = 7^(-1).
The expression simplifies to 7^(-1) / 6^(-2).
Step 3: Apply the negative exponent property a^(-m) = 1/a^m.
7^(-1) = 1/7^1 = 1/7.
The expression becomes 1/7 / 6^(-2).
Step 4: Apply the division of powers property a^m / a^n = a^(m-n).
1/7 / 6^(-2) = 1/7 / 6^2 = 1/7 / 36.
Now, divide 1 by 7 and express the result as an improper fraction:
1 ÷ 7 = 1/7.
Therefore, the simplified expression is 1/7 / 36, and the answer as an improper fraction is (1/7) / 36 = 1/252.
you sure?
GPT 3.5
I apologize for the confusion once again. I made a mistake in my previous response. Let's correct it.
We have the expression (7^(-3))^7 ⋅ 7^20/6^(-2).
Let's simplify each part separately.
(7^(-3))^7 can be simplified using the property (a^m)^n = a^(m*n).
(7^(-3))^7 = 7^((-3)*7) = 7^(-21).
The expression now becomes 7^(-21) ⋅ 7^20/6^(-2).
Now, let's simplify the powers of 7.
7^(-21) ⋅ 7^20 = 7^(-21+20) = 7^(-1).
The expression now becomes 7^(-1) / 6^(-2).
Next, let's apply the property a^(-m) = 1/a^m to simplify the negative exponents.
7^(-1) = 1/7^1 = 1/7.
The expression now becomes 1/7 / 6^(-2).
To simplify 6^(-2), we use the property a^(-m) = 1/a^m.
6^(-2) = 1/6^2 = 1/36.
The expression now becomes 1/7 / 1/36.
Dividing by a fraction is the same as multiplying by its reciprocal.
1/7 / 1/36 = 1/7 * 36/1 = (1 * 36) / (7 * 1) = 36/7.
Therefore, the simplified expression is 36/7, where 36 is the numerator and 7 is the denominator.
We have the expression (7^(-3))^7 ⋅ 7^20/6^(-2).
Let's simplify each part separately.
(7^(-3))^7 can be simplified using the property (a^m)^n = a^(m*n).
(7^(-3))^7 = 7^((-3)*7) = 7^(-21).
The expression now becomes 7^(-21) ⋅ 7^20/6^(-2).
Now, let's simplify the powers of 7.
7^(-21) ⋅ 7^20 = 7^(-21+20) = 7^(-1).
The expression now becomes 7^(-1) / 6^(-2).
Next, let's apply the property a^(-m) = 1/a^m to simplify the negative exponents.
7^(-1) = 1/7^1 = 1/7.
The expression now becomes 1/7 / 6^(-2).
To simplify 6^(-2), we use the property a^(-m) = 1/a^m.
6^(-2) = 1/6^2 = 1/36.
The expression now becomes 1/7 / 1/36.
Dividing by a fraction is the same as multiplying by its reciprocal.
1/7 / 1/36 = 1/7 * 36/1 = (1 * 36) / (7 * 1) = 36/7.
Therefore, the simplified expression is 36/7, where 36 is the numerator and 7 is the denominator.
Apply the properties of integer exponents to generate equivalent expression to (7^-3)^7 times 7^20/6^-2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.
GPT 3.5
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents:
1) (a^m)^n = a^(m*n)
2) a^(-m) = 1/a^m
Let's simplify the expression step by step:
First, let's simplify (7^(-3))^7. Applying property 1:
(7^(-3))^7 = 7^((-3)*7) = 7^(-21)
Next, let's simplify 7^20 and 6^(-2):
7^20/6^(-2) = 7^20/(1/6^2) = 7^20/6^2
Finally, let's combine the simplified expressions:
(7^(-3))^7 ⋅ 7^20/6^(-2) = 7^(-21) ⋅ 7^20/6^2
Now, we apply property 2 to convert the negative exponents:
7^(-21) = 1/7^21 and 6^2 = 36
We have:
(1/7^21) ⋅ 7^20/36
To simplify, we can multiply the numerator and denominator together:
(1/7^21) ⋅ 7^20 = 7^(20-21) = 7^(-1) = 1/7
Putting it all together:
(1/7^21) ⋅ 7^20/36 = (1/7) / 36
Dividing by a fraction is the same as multiplying by its reciprocal, so we can rewrite as:
(1/7) / 36 = (1/7) * (1/36) = 1/7 * 1/36 = 1/252
Therefore, the simplified expression is 1/252, where 1 is the numerator and 252 is the denominator.
1) (a^m)^n = a^(m*n)
2) a^(-m) = 1/a^m
Let's simplify the expression step by step:
First, let's simplify (7^(-3))^7. Applying property 1:
(7^(-3))^7 = 7^((-3)*7) = 7^(-21)
Next, let's simplify 7^20 and 6^(-2):
7^20/6^(-2) = 7^20/(1/6^2) = 7^20/6^2
Finally, let's combine the simplified expressions:
(7^(-3))^7 ⋅ 7^20/6^(-2) = 7^(-21) ⋅ 7^20/6^2
Now, we apply property 2 to convert the negative exponents:
7^(-21) = 1/7^21 and 6^2 = 36
We have:
(1/7^21) ⋅ 7^20/36
To simplify, we can multiply the numerator and denominator together:
(1/7^21) ⋅ 7^20 = 7^(20-21) = 7^(-1) = 1/7
Putting it all together:
(1/7^21) ⋅ 7^20/36 = (1/7) / 36
Dividing by a fraction is the same as multiplying by its reciprocal, so we can rewrite as:
(1/7) / 36 = (1/7) * (1/36) = 1/7 * 1/36 = 1/252
Therefore, the simplified expression is 1/252, where 1 is the numerator and 252 is the denominator.