Apply the Properties of Integer Exponents to generate equivalent expressions to 3^7⋅3^−9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
Apply the Properties of Integer Exponents to generate equivalent expressions to 2^2/2^−4 with only positive exponents, then solve the expression. Provide the solution as a whole number.
Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the property of integer exponents that states:
For any nonzero number a, a^0 = 1.
Using this property, we can rewrite the expression 4^0 as 1.
To rewrite 5^−2 with only positive exponents, we can use another property of integer exponents that states:
For any nonzero number a, a^(-n) = 1/a^n.
Applying this property, we can rewrite 5^−2 as 1/5^2.
To rewrite (−2)^−3 with only positive exponents, we can use the reciprocal property of integer exponents that states:
For any nonzero number a^(-n), 1/a^(-n) = a^n.
Applying this property, we can rewrite (−2)^−3 as 1/(-2)^3.
Now let's simplify the expression:
1 ⋅ (1/5^2) / (1/(-2)^3)
Simplifying further:
1 ⋅ 1/(5^2) ÷ 1/(-2)^3
Since division is the same as multiplying by the reciprocal of the denominator, we can rewrite the expression as:
1 ⋅ 1/(5^2) ⋅ ((-2)^3/1)
Simplifying further:
1/(5^2) ⋅ (-2)^3
Now let's simplify the exponents:
1/25 ⋅ (-8)
Finally, calculating the product:
1/25 ⋅ (-8) = -8/25
Therefore, the expression 4^0 ⋅ 5^−2/(−2)^−3 is equivalent to -8/25.
Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27^−7)^0/3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. For any nonzero number a, a^0 = 1.
2. For any nonzero numbers a and b, (a^m)^n = a^(m * n).
3. For any nonzero numbers a and b, (a * b)^n = a^n * b^n.
4. For any nonzero numbers a and b, (a / b)^n = a^n / b^n.
Now, let's apply these properties to the expression (14^3⋅27^−7)^0/3^4:
First, let's simplify the exponents within the parentheses using properties 1, 2, and 4:
(14^3⋅27^−7)^0 = (14^3)^0 * (27^−7)^0 = 1 * 1 = 1.
Next, let's simplify the denominator:
3^4 = 81.
Now, the expression becomes:
1/81.
Therefore, the expression (14^3⋅27^−7)^0/3^4 can be simplified to 1/81.
Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the property of integer exponents that states:
For any nonzero number a and integers m and n, a^m ⋅ a^n = a^(m + n)
Using this property, we can rewrite the expression 3^7⋅3^−9 as:
3^(7 + (-9))
Simplifying the exponent:
3^(7 + (-9)) = 3^(-2)
To further simplify the expression with a negative exponent, we can use another property of integer exponents that states:
For any nonzero number a and integer n, a^(-n) = 1/a^n
Applying this property:
3^(-2) = 1/3^2
Simplifying the fraction:
1/3^2 = 1/9
Therefore, the expression 3^7⋅3^−9 is equivalent to 1/9.
To generate equivalent expressions with only positive exponents, we can use the property of integer exponents that states:
For any nonzero number a and integers m and n, a^m / a^n = a^(m - n).
Using this property, we can rewrite the expression 2^2/2^−4 as:
2^(2 - (-4))
Simplifying the exponent:
2^(2 - (-4)) = 2^(2 + 4)
Simplifying further:
2^(2 + 4) = 2^6
Therefore, the expression 2^2/2^−4 is equivalent to 2^6.
Solving the expression:
2^6 = 64
So, the solution to the expression 2^2/2^−4 is 64.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. For any nonzero number a, a^0 = 1.
2. For any nonzero numbers a and b, (a^m)^n = a^(m * n).
3. For any nonzero numbers a and b, (a * b)^n = a^n * b^n.
4. For any nonzero numbers a and b, (a / b)^n = a^n / b^n.
5. For any nonzero number a, a^(-n) = 1/a^n.
To simplify the expression (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13, let's break it down step by step:
First, let's simplify the exponents within the parentheses using property 5:
(4^3 ⋅ 7^−2)^−5 = (4^3 ⋅ 1/7^2)^−5 = (4^3/7^2)^−5.
Now, let's simplify the denominator using property 4:
(4^3/7^2)^−5 = (4^3)^−5 / (7^2)^−5.
Now, let's simplify the exponents using property 2:
(4^3)^−5 / (7^2)^−5 = 4^(3 * -5) / 7^(2 * -5).
Simplifying further:
4^(3 * -5) = 4^(-15) = 1/4^15.
7^(2 * -5) = 7^(-10) = 1/7^10.
Now, let's substitute these values back into the expression:
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 = (1/4^15) / (1/7^10) ⋅ 4^(-13).
Applying properties 3 and 4 to simplify the expression further:
(1/4^15) / (1/7^10) ⋅ 4^(-13) = (1/4^15) ⋅ (7^10) / 4^13.
Using property 4 again to combine the fractions:
(1/4^15) ⋅ (7^10) / 4^13 = 7^10 / 4^(15 - 13).
Simplifying the exponent:
4^(15 - 13) = 4^2 = 16.
Now, substitute back into the expression:
7^10 / 16.
Therefore, the expression (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 can be simplified to 7^10 / 16.
are you sure?
I apologize for the mistake in my previous response. Let's revisit the problem and solve it correctly.
To simplify the expression (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, we'll apply the property of negative exponents and simplify each part:
First, let's simplify the base inside the parentheses using the property of negative exponents:
(4^3 ⋅ 7^−2)^−5 = (4^3 / 7^2)^−5
Next, let's simplify the denominator using the property of negative exponents:
(4^3 / 7^2)^−5 = (4^3)^−5 / (7^2)^−5
Now, let's simplify the exponents:
(4^3)^−5 / (7^2)^−5 = 4^(3 * -5) / 7^(2 * -5)
Simplifying further:
4^(3 * -5) = 4^(-15) = 1/4^15
7^(2 * -5) = 7^(-10) = 1/7^10
Now, let's substitute these values back into the expression:
(1/4^15) / (1/7^10) ⋅ 4^−13
Applying the property of negative exponents and simplifying the expression further:
(1/4^15) / (1/7^10) ⋅ 4^−13 = (7^10 / 4^15) ⋅ 4^−13
Using the property of exponent addition:
(7^10 / 4^15) ⋅ 4^−13 = 7^10 / (4^15 ⋅ 4^13)
Applying the property of exponent addition again:
7^10 / (4^15 ⋅ 4^13) = 7^10 / 4^(15+13) = 7^10 / 4^28
Therefore, the simplified expression is 7^10 / 4^28.
thats not right
Apologies for the incorrect response. Let's solve the expression correctly:
To simplify the expression (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, we can simplify each part separately.
First, let's simplify the base inside the parentheses:
4^3 ⋅ 7^−2 = 64 ⋅ 1/49 = 64/49
Next, let's simplify the denominator:
7^9 = 49^5
Now, let's simplify the exponent of 4^−13:
4^−13 = 1/4^13
Substituting these values back into the expression:
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 = (64/49)^−5/49^5 ⋅ 1/4^13
Using the property of negative exponents:
(64/49)^−5/49^5 ⋅ 1/4^13 = (49/64)^5/49^5 ⋅ 1/4^13
Now, let's simplify the exponent:
(49/64)^5 = 49^5/64^5
Substituting this back into the expression:
(49^5/64^5)/49^5 ⋅ 1/4^13 = (49^5/64^5)/(49^5 ⋅ 1/4^13)
Simplifying further:
(49^5/64^5)/(49^5 ⋅ 1/4^13) = 49^5/(64^5 ⋅ 49^5 ⋅ 1/4^13)
Applying the property of exponent subtraction:
49^5/(64^5 ⋅ 49^5 ⋅ 1/4^13) = 49^5/[(64 ⋅ 49)^5 ⋅ 1/4^13]
Simplifying further:
49^5/[(64 ⋅ 49)^5 ⋅ 1/4^13] = 49^5/(3136^5 ⋅ 1/4^13)
Finally, let's simplify 3136^5 to 961729203785216:
49^5/(3136^5 ⋅ 1/4^13) = 49^5/(961729203785216 ⋅ 1/4^13)
Therefore, the expression simplifies to 49^5/(961729203785216 ⋅ 1/4^13).
Which of the following is an equivalent expression to 5^9⋅5^−13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)
Responses
1/5^4
1/625
59/5^−13
5^4
1/5^4 is the correct equivalent expression generated by applying the Properties of Integer Exponents.
Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9? (1 point)
Responses
Only the Product Rule of Exponents
Only the Power Rule of Exponents
Only the Property of Negative Exponents
The Property of Negative Exponents and the Product Rule of Exponents
The Property of Negative Exponents and the Product Rule of Exponents were used to generate the equivalent expression 3^14 from 3^5/3^−9.
Which of the following is an equivalent expression to 15^0 ⋅ 7^−2/(−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)
Responses
4^3/7^2
1/7^2 ⋅ (−4)^−3
(−4)^3/7^2
7^2/(−4)^3
The equivalent expression that has been generated by applying the Properties of Integer Exponents is 7^2/(−4)^3.
Which of the following is the equivalent expression to (15^−3⋅ 4^7)0^/4^−3 that has been generated by applying the Properties of Integer Exponents? (1 point)
Responses
64
1⋅4^−3
1/64
60/4^−3
The equivalent expression that has been generated by applying the Properties of Integer Exponents is 1/64.
Which of the following is an equivalent expression to (2^−3 ⋅9^3)^−4/2^9⋅9^−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
Responses
2^12⋅9^−12/2^9⋅9^−10
2^−7⋅9^−1/2^9⋅9^−10
2^3/9^2
2^3 ⋅9^2