To apply the properties of integer exponents, we can use the following rules:
1. To simplify a negative exponent, we can take the reciprocal of the base and make the exponent positive.
2. When multiplying two expressions with the same base, we can add their exponents.
3. When dividing two expressions with the same base, we can subtract their exponents.
Using these rules, let's simplify the given expression step by step:
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13
Step 1: Apply the negative exponent rule to (7^-2), making it (1/7^2):
(4^3 ⋅ (1/7^2))^-5/7^9 ⋅ 4^-13
Step 2: Multiply the exponents inside the parentheses (4^3 and 1/7^2):
4^(3 * -5)/7^(2 * -5) ⋅ 4^-13
Simplifying the exponents:
4^-15/7^-10 ⋅ 4^-13
Step 3: Apply the negative exponent rule to (4^-15), making it (1/4^15):
(1/4^15)/(7^-10) ⋅ 4^-13
Step 4: Divide the exponents (1/4^15 divided by 7^-10):
1/(4^15 * 7^-10) ⋅ 4^-13
Step 5: Apply the negative exponent rule to (7^-10), making it (1/7^10):
1/(4^15 * (1/7^10)) ⋅ 4^-13
Step 6: Multiply the expressions:
1/(4^15/7^10) ⋅ 4^-13
Step 7: Simplify 4^15/7^10 by subtracting the exponents:
1/(4^15/7^10) ⋅ 4^-13 = 1/((4^15)/(7^10)) ⋅ 4^-13
Step 8: Apply the negative exponent rule to (4^-13), making it (1/4^13):
1/((4^15)/(7^10)) ⋅ (1/4^13)
Step 9: Multiply the expressions:
1 * (1/((4^15)/(7^10) * 1/4^13)
Simplifying the multiplication:
1/(4^15/7^10 * 1/4^13)
Step 10: Multiply the fractions:
1/(4^15 * 7^10/4^13)
Step 11: Divide the exponents by subtracting them:
1/(4^(15-13) * 7^10)
Step 12: Simplify the exponents:
1/(4^2 * 7^10)
Step 13: Evaluate 4^2 and 7^10:
1/(16 * 282475249)
Step 14: Simplify the division:
1/4519603984
Therefore, (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 = 1/4519603984.
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.(1 point)
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 =
17 answers
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13=
To simplify the given expression, let's break it down step by step:
First, we can simplify the expression within the parentheses:
4^3 ⋅ 7^(-2) = 64 ⋅ 1/49 = 64/49
Now, let's substitute the simplified expression back into the original one:
(64/49)^(-5/7^9) ⋅ 4^(-13)
Next, we can apply the rule of negative exponents to the base 64/49:
(49/64)^(5/7^9) ⋅ 4^(-13)
Now, we can simplify the expression 4^(-13) using the rule of negative exponents:
(49/64)^(5/7^9) ⋅ 1/4^13
Lastly, we can combine the exponents of 49/64:
(49/64)^(5/7^9) ⋅ 1/2^26
So, the equivalent expression to (4^3 ⋅ 7^(-2))^(-5/7^9) ⋅ 4^(-13) is (49/64)^(5/7^9) ⋅ 1/2^26.
First, we can simplify the expression within the parentheses:
4^3 ⋅ 7^(-2) = 64 ⋅ 1/49 = 64/49
Now, let's substitute the simplified expression back into the original one:
(64/49)^(-5/7^9) ⋅ 4^(-13)
Next, we can apply the rule of negative exponents to the base 64/49:
(49/64)^(5/7^9) ⋅ 4^(-13)
Now, we can simplify the expression 4^(-13) using the rule of negative exponents:
(49/64)^(5/7^9) ⋅ 1/4^13
Lastly, we can combine the exponents of 49/64:
(49/64)^(5/7^9) ⋅ 1/2^26
So, the equivalent expression to (4^3 ⋅ 7^(-2))^(-5/7^9) ⋅ 4^(-13) is (49/64)^(5/7^9) ⋅ 1/2^26.
Which of the following is an equivalent expression to 59⋅5−13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)
Responses
54
5 superscript 4 baseline
154
Start Fraction 1 over 5 superscript 4 baseline end fraction
1625
Start Fraction 1 over 625 end fraction
595−13
Responses
54
5 superscript 4 baseline
154
Start Fraction 1 over 5 superscript 4 baseline end fraction
1625
Start Fraction 1 over 625 end fraction
595−13
To generate an equivalent expression with only positive exponents, we can apply the property of negative exponents, which states that when we have a negative exponent, we can rewrite it as the reciprocal with a positive exponent. However, it seems that the expression given, 59⋅5−13, already only has positive exponents. Therefore, none of the given responses is an equivalent expression generated by applying the Properties of Integer Exponents.
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
a. 54
5 superscript 4 baseline
b. 154
Start Fraction 1 over 5 superscript 4 baseline end fraction
c. 1625
Start Fraction 1 over 625 end fraction
d. 595−13
Responses
a. 54
5 superscript 4 baseline
b. 154
Start Fraction 1 over 5 superscript 4 baseline end fraction
c. 1625
Start Fraction 1 over 625 end fraction
d. 595−13
To find an equivalent expression with only positive exponents, we can use the property of product of exponents, which states that when we multiply two expressions with the same base, we can add their exponents.
In the given expression, 5^9 * 5^(-13), we have the same base of 5. So, we can add the exponents:
5^(9 + (-13))
Simplifying the exponent gives us:
5^(-4)
Now, we can apply the property of negative exponents, which states that when we have a negative exponent, we can rewrite it as the reciprocal with a positive exponent:
1 / 5^4
Therefore, the equivalent expression to 5^9 * 5^(-13) with only positive exponents is:
1 / 5^4
Looking at the given responses:
a. 54
b. 154
c. 1625
d. 595^(-13)
None of the responses match the equivalent expression we found, which is 1 / 5^4.
In the given expression, 5^9 * 5^(-13), we have the same base of 5. So, we can add the exponents:
5^(9 + (-13))
Simplifying the exponent gives us:
5^(-4)
Now, we can apply the property of negative exponents, which states that when we have a negative exponent, we can rewrite it as the reciprocal with a positive exponent:
1 / 5^4
Therefore, the equivalent expression to 5^9 * 5^(-13) with only positive exponents is:
1 / 5^4
Looking at the given responses:
a. 54
b. 154
c. 1625
d. 595^(-13)
None of the responses match the equivalent expression we found, which is 1 / 5^4.
wrong which one is the answer
Apologies for the confusion. Let's go through the options again and find the correct equivalent expression.
The original expression is 5^9 * 5^(-13). To find the equivalent expression with only positive exponents, we can use the property of product of exponents, which states that when we multiply two expressions with the same base, we can add their exponents.
In this case, we have:
5^9 * 5^(-13) = 5^(9 + (-13)) = 5^(-4)
Now, let's simplify the negative exponent. The property of negative exponents states that a negative exponent can be rewritten as the reciprocal with a positive exponent.
Therefore, 5^(-4) can be rewritten as:
1 / 5^4
Looking again at the given responses:
a. 54
b. 154
c. 1625
d. 595^(-13)
The correct answer is:
c. Start Fraction 1 over 625 end fraction
The equivalent expression to 5^9 * 5^(-13) with only positive exponents that has been generated by applying the Properties of Integer Exponents is Start Fraction 1 over 625 end fraction.
The original expression is 5^9 * 5^(-13). To find the equivalent expression with only positive exponents, we can use the property of product of exponents, which states that when we multiply two expressions with the same base, we can add their exponents.
In this case, we have:
5^9 * 5^(-13) = 5^(9 + (-13)) = 5^(-4)
Now, let's simplify the negative exponent. The property of negative exponents states that a negative exponent can be rewritten as the reciprocal with a positive exponent.
Therefore, 5^(-4) can be rewritten as:
1 / 5^4
Looking again at the given responses:
a. 54
b. 154
c. 1625
d. 595^(-13)
The correct answer is:
c. Start Fraction 1 over 625 end fraction
The equivalent expression to 5^9 * 5^(-13) with only positive exponents that has been generated by applying the Properties of Integer Exponents is Start Fraction 1 over 625 end fraction.
Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?(1 point)
Responses
only the Property of Negative Exponents
only the Property of Negative Exponents
only the Product Rule of Exponents
only the Product Rule of Exponents
the Property of Negative Exponents and the Product Rule of Exponents
the Property of Negative Exponents and the Product Rule of Exponents
only the Power Rule of Exponents
Responses
only the Property of Negative Exponents
only the Property of Negative Exponents
only the Product Rule of Exponents
only the Product Rule of Exponents
the Property of Negative Exponents and the Product Rule of Exponents
the Property of Negative Exponents and the Product Rule of Exponents
only the Power Rule of Exponents
The equivalent expression was generated using the property of negative exponents.
In the original expression, 3^5/3^(-9), there is a negative exponent in the denominator. The property of negative exponents states that when we have a negative exponent, we can rewrite it as the reciprocal with a positive exponent.
Applying this property to the expression gives us:
3^5 * 3^9
Since two expressions with the same base are being multiplied, we can apply the product rule of exponents, which states that when we multiply two expressions with the same base, we can add their exponents.
Therefore, the equivalent expression 3^14 was generated using only the property of negative exponents.
The correct response is:
only the Property of Negative Exponents
In the original expression, 3^5/3^(-9), there is a negative exponent in the denominator. The property of negative exponents states that when we have a negative exponent, we can rewrite it as the reciprocal with a positive exponent.
Applying this property to the expression gives us:
3^5 * 3^9
Since two expressions with the same base are being multiplied, we can apply the product rule of exponents, which states that when we multiply two expressions with the same base, we can add their exponents.
Therefore, the equivalent expression 3^14 was generated using only the property of negative exponents.
The correct response is:
only the Property of Negative Exponents
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
72(−4)3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction
172 ⋅ (−4)−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction
(−4)372
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction
4372
Responses
72(−4)3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction
172 ⋅ (−4)−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction
(−4)372
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction
4372
To find an equivalent expression with only positive exponents, we can apply the property of negative exponents, which states that a negative exponent can be rewritten as the reciprocal with a positive exponent. Additionally, any number raised to the power of 0 is equal to 1.
Let's simplify the given expression step by step:
1) 15^0 = 1 (any number raised to the power of 0 is equal to 1)
2) 7^(-2) = 1 / 7^2 (applying the property of negative exponents)
3) (-4)^(-3) = 1 / (-4)^3 (applying the property of negative exponents)
Now, we substitute these simplifications back into the original expression:
1 * (1 / 7^2) / (1 / (-4)^3)
Simplifying further:
1 * (1 / 7^2) * (-4)^3
Since raising -4 to the power of 3 is positive, we can leave it as is:
1 * (1 / 7^2) * (-4)^3
The result is:
(1 / 7^2) * (-4)^3
Comparing this result to the given responses:
1) 72(-4)^3
2) Start Fraction 7^2 / (-4)^3
3) (−4)^3 / 7^2
4) 4372
The correct answer is:
2) Start Fraction 7^2 / (-4)^3
The equivalent expression to 15^0 * 7^(-2) / (-4)^(-3) with only positive exponents that has been generated by applying the Properties of Integer Exponents is Start Fraction 7^2 / (-4)^3.
Let's simplify the given expression step by step:
1) 15^0 = 1 (any number raised to the power of 0 is equal to 1)
2) 7^(-2) = 1 / 7^2 (applying the property of negative exponents)
3) (-4)^(-3) = 1 / (-4)^3 (applying the property of negative exponents)
Now, we substitute these simplifications back into the original expression:
1 * (1 / 7^2) / (1 / (-4)^3)
Simplifying further:
1 * (1 / 7^2) * (-4)^3
Since raising -4 to the power of 3 is positive, we can leave it as is:
1 * (1 / 7^2) * (-4)^3
The result is:
(1 / 7^2) * (-4)^3
Comparing this result to the given responses:
1) 72(-4)^3
2) Start Fraction 7^2 / (-4)^3
3) (−4)^3 / 7^2
4) 4372
The correct answer is:
2) Start Fraction 7^2 / (-4)^3
The equivalent expression to 15^0 * 7^(-2) / (-4)^(-3) with only positive exponents that has been generated by applying the Properties of Integer Exponents is Start Fraction 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
64
604−3
Start Fraction 60 over 4 negative cubed end fraction
164
Start Fraction 1 over 64 end fraction
1⋅4−3
Responses
64
64
604−3
Start Fraction 60 over 4 negative cubed end fraction
164
Start Fraction 1 over 64 end fraction
1⋅4−3
To find the equivalent expression, let's simplify step by step while applying the properties of integer exponents:
1. Distribute the exponent of 0 to each term within the parentheses:
(15^-3 * 4^7)^0 = 1^0 = 1
2. Rewrite 4^-3 using the property of negative exponents:
4^-3 = 1/4^3
Now, substitute the simplified expressions back into the original expression:
1 / 4^3 = 1 / 64
Looking at the given options:
1) 64
2) Start Fraction 60 / 4^(-3)
3) Start Fraction 1 / 64
4) 1 * 4^(-3)
The correct answer is:
3) Start Fraction 1 / 64
This is the equivalent expression to (15^(-3) * 4^7)^0 / 4^(-3) that has been generated by applying the Properties of Integer Exponents.
1. Distribute the exponent of 0 to each term within the parentheses:
(15^-3 * 4^7)^0 = 1^0 = 1
2. Rewrite 4^-3 using the property of negative exponents:
4^-3 = 1/4^3
Now, substitute the simplified expressions back into the original expression:
1 / 4^3 = 1 / 64
Looking at the given options:
1) 64
2) Start Fraction 60 / 4^(-3)
3) Start Fraction 1 / 64
4) 1 * 4^(-3)
The correct answer is:
3) Start Fraction 1 / 64
This is the equivalent expression to (15^(-3) * 4^7)^0 / 4^(-3) that has been generated by applying the Properties of Integer Exponents.
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
212⋅9−1229⋅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
2−7⋅9−129⋅9−10
Start fraction 2 superscript negative 7 baseline times 9 superscript negative 1 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction
2392
Start Fraction 2 cubed over 9 squared end fraction
23 ⋅92
Responses
212⋅9−1229⋅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
2−7⋅9−129⋅9−10
Start fraction 2 superscript negative 7 baseline times 9 superscript negative 1 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction
2392
Start Fraction 2 cubed over 9 squared end fraction
23 ⋅92
To find the equivalent expression with only positive exponents, let's simplify the given expression step by step:
1. Apply the property of negative exponents to the base 2 and 9:
(2^-3 * 9^3)^-4 / 2^9 * 9^-10 = (1/2^3 * 9^3)^-4 / 2^9 * 1/9^10
2. Simplify the expressions within the parentheses using the property of negative exponents:
(1/8 * 9^3)^-4 / 2^9 * 1/9^10 = (9^3/8)^-4 / 2^9 * 1/9^10
3. Combine the exponents by applying the property of negative exponents and the product rule of exponents:
(9^(3*-4) / 8^-4) / 2^9 * 1/9^10 = (9^-12 / 8^-4) / 2^9 * 1/9^10
4. Simplify the negative exponents using the property of negative exponents:
(8^4/9^12) / 2^9 * 1/9^10
5. Combine the fractions by subtracting the exponents:
8^4 / 9^12 / (2^9 * 9^10)
6. Combine the bases with the same exponent:
8^4 / (2^9 * 9^22)
Therefore, the equivalent expression to (2^-3 * 9^3)^-4 / 2^9 * 9^-10 with only positive exponents is:
8^4 / (2^9 * 9^22)
Looking at the given responses:
1) Start fraction 2^12 * 9^-12 / 2^9 * 9^-10
2) Start fraction 2^-7 * 9^-1 / 2^9 * 9^-10
3) Start fraction 2^3 / 9^2
4) 2^3 * 9^2
The correct answer is:
3) Start fraction 2^3 / 9^2
This is the 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. Apply the property of negative exponents to the base 2 and 9:
(2^-3 * 9^3)^-4 / 2^9 * 9^-10 = (1/2^3 * 9^3)^-4 / 2^9 * 1/9^10
2. Simplify the expressions within the parentheses using the property of negative exponents:
(1/8 * 9^3)^-4 / 2^9 * 1/9^10 = (9^3/8)^-4 / 2^9 * 1/9^10
3. Combine the exponents by applying the property of negative exponents and the product rule of exponents:
(9^(3*-4) / 8^-4) / 2^9 * 1/9^10 = (9^-12 / 8^-4) / 2^9 * 1/9^10
4. Simplify the negative exponents using the property of negative exponents:
(8^4/9^12) / 2^9 * 1/9^10
5. Combine the fractions by subtracting the exponents:
8^4 / 9^12 / (2^9 * 9^10)
6. Combine the bases with the same exponent:
8^4 / (2^9 * 9^22)
Therefore, the equivalent expression to (2^-3 * 9^3)^-4 / 2^9 * 9^-10 with only positive exponents is:
8^4 / (2^9 * 9^22)
Looking at the given responses:
1) Start fraction 2^12 * 9^-12 / 2^9 * 9^-10
2) Start fraction 2^-7 * 9^-1 / 2^9 * 9^-10
3) Start fraction 2^3 / 9^2
4) 2^3 * 9^2
The correct answer is:
3) Start fraction 2^3 / 9^2
This is the equivalent expression to (2^-3 * 9^3)^-4 / 2^9 * 9^-10 with only positive exponents, generated by applying the Properties of Integer Exponents.