Apply All Exponent Rules Quick Check
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Question
Which of the following is the equivalent expression to (15^−3 ⋅ 4^7)^0over4^−3 that has been generated by applying the Properties of Integer Exponents?(1 point)
Responses
1⋅4^−3
1 times 4 negative cubed
64
1over64
Start Fraction 1 over 64 end fraction
60over4^−3
To simplify the expression (15^−3 ⋅ 4^7)^0over4^−3, we can apply the exponent rules:
First, let's simplify the exponents separately:
15^−3 = 1/(15^3) = 1/3375
4^7 = 16384
Now, let's simplify the expression inside the parentheses:
(1/3375 ⋅ 16384)^0 = 1^0 = 1
Finally, let's simplify the denominator of the exponent:
4^−3 = 1/(4^3) = 1/64
Putting it all together:
(15^−3 ⋅ 4^7)^0over4^−3 = 1^0/1/64 = 1/(1/64) = 64
Therefore, the equivalent expression is 64.
Which of the following is an equivalent expression to (2^−3⋅9^3)^−4over2^9 ⋅9^−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
Responses
2^3over9^2
Start Fraction 2 cubed over 9 squared end fraction
2^12⋅9^−12over2^9⋅9^−10
Start fraction 2 superscript 12 baseline times 9 superscript negative 12 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction
2^3 ⋅9^2
2 cubed times 9 squared
2^−7⋅9^−1over2^9⋅9^−10
To simplify the expression (2^−3⋅9^3)^−4over2^9 ⋅9^−10 with only positive exponents, we can apply the exponent rules:
First, let's simplify the exponents separately:
2^−3 = 1/(2^3) = 1/8
9^3 = 729
Now, let's simplify the expression inside the parentheses:
(1/8 ⋅ 729)^−4 = (729/8)^−4
Next, let's simplify the numerator and denominator of the second fraction:
2^9 = 512
9^−10 = 1/(9^10) = 1/3486784401
Putting it all together:
(2^−3⋅9^3)^−4over2^9 ⋅9^−10 = (729/8)^−4/512⋅(1/3486784401) = (8/729)^4/512⋅(1/3486784401) = (8^4/729^4)/(512/3486784401)
Simplifying further:
(8^4/729^4)/(512/3486784401) = (4096/531441)/(512/3486784401) = (4096/531441) × (3486784401/512) = 2^12/9^2
Therefore, the equivalent expression is 2^12/9^2.
Are these correct?
Yes, both of the answer choices you provided are correct. Great job!
1. 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)
5^4
5^9/5^−13
1/625
1/5^4
2. Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?(1 point)
only the Property of Negative Exponents
only the Power Rule of Exponents
the Property of Negative Exponents and the Product Rule of Exponents
only the Product Rule of Exponents
3. 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)
1/7^2 ⋅ (−4)^−3
4^3/7^2
(−4)^3/7^2
7^2/(−4)^3
4. 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)
1/64
60/4^−3
64
1⋅4^−3
5. Which of the following is an equivalent expression to (2−3 ⋅93)−429 ⋅9−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
2^3 ⋅ 9^2
2^12 ⋅ 9^−12/2^9⋅9^−10
2^3/9^2
2^−7⋅9^−1/ 2^9⋅9^−10
Let's go through each question one by one:
1. 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?
The correct answer is 5^4. (Using the product rule of exponents, 5^9 ⋅ 5^−13 = 5^(9-13) = 5^(-4) = 1/5^4)
2. Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?
The correct answer is only the Property of Negative Exponents. (Using the Quotient Rule of exponents, 3^5/3^−9 = 3^(5-(-9)) = 3^14)
3. 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?
The correct answer is 4^3/7^2. (Using the property 15^0 = 1, (-4)^-3 = 1/(-4)^3, and applying the quotient rule of exponents)
4. 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?
The correct answer is 64. (Using the property (a^b)^c = a^(b*c), 15^(-3 * 0) = 15^0 = 1, and 4^(-3 * 0) = 4^0 = 1)
5. Which of the following is an equivalent expression to (2^−3 ⋅9^3)−429 ⋅9^−10 with only positive exponents, generated by applying the Properties of Integer Exponents?
The correct answer is 2^3 ⋅ 9^2. (Using the property (a^b)^c = a^(b*c), and simplifying the exponents)
Only number 4 was correct..
I apologize for the incorrect answers. Let's go through the questions again and provide the correct responses:
1. 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?
The correct answer is 5^(-4). (Applying the product rule of exponents, 5^9 ⋅ 5^(-13) = 5^(9-13) = 5^(-4))
2. Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?
The correct answer is only the Product Rule of Exponents. (Using the quotient rule of exponents, 3^5/3^(-9) = 3^(5-(-9)) = 3^14)
3. 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?
The correct answer is 7^2/(−4)^3. (Using the property 15^0 = 1, and simplifying the exponents)
5. 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?
The correct answer is 2^12 ⋅ 9^(-12)/2^9 ⋅ 9^(-10). (Applying the negative exponent rule)
Once again, I apologize for the previous incorrect responses.
To find the equivalent expression generated by applying the properties of integer exponents, let's break down the given expression step by step and simplify it.
Step 1: Start with the given expression: (15^−3 ⋅ 4^7)^0 / 4^−3
Step 2: Apply the exponent rule for a negative exponent, which states that a^−n = 1 / a^n. We have two negative exponents in the expression: 15^−3 and 4^−3.
15^−3 = 1 / 15^3
4^−3 = 1 / 4^3
Now the expression becomes:
(1 / 15^3 ⋅ 4^7)^0 / (1 / 4^3)
Step 3: Apply the exponent rule for a zero exponent, which states that a^0 = 1. Since the entire expression is raised to the power of 0, it will simplify to 1.
(1 / 15^3 ⋅ 4^7)^0 / (1 / 4^3) = 1 / (1 / 4^3)
Step 4: Divide by a fraction is the same as multiplying by its reciprocal. So, we can rewrite 1 / (1 / 4^3) as 1 * (4^3 / 1).
1 / (1 / 4^3) = 1 * (4^3 / 1) = 4^3
Step 5: Simplify 4^3, which means multiplying 4 by itself three times.
4^3 = 4 * 4 * 4 = 64
Therefore, the equivalent expression generated by applying the properties of integer exponents is 64.
So, the correct response is: 64.