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25 answers
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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.(1 point)
To generate equivalent expressions with only positive exponents, we can use the property of integer exponents that states:
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, we get:
3^(-2)
To solve the expression 3^(-2), we can use another property of integer exponents that states:
a^(-n) = 1 / a^n
Applying this property, we have:
3^(-2) = 1 / 3^2
Simplifying further, we get:
3^(-2) = 1 / 9
Therefore, the simplified fraction with no remaining exponents is 1/9.
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, we get:
3^(-2)
To solve the expression 3^(-2), we can use another property of integer exponents that states:
a^(-n) = 1 / a^n
Applying this property, we have:
3^(-2) = 1 / 3^2
Simplifying further, we get:
3^(-2) = 1 / 9
Therefore, the simplified fraction with no remaining exponents is 1/9.
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.(1 point)
To generate equivalent expressions with only positive exponents, we can use the property of integer exponents that states:
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 exponents, we get:
2^(2 + 4)
2^6
Therefore, the solution to the expression 2^2 / 2^(-4) with only positive exponents is 2^6, which equals 64.
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 exponents, we get:
2^(2 + 4)
2^6
Therefore, the solution to the expression 2^2 / 2^(-4) with only positive exponents is 2^6, which equals 64.
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.(1 point)
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents. Let's simplify the expression step by step:
First, recall that any number (except 0) raised to the power of 0 is equal to 1. Therefore, we have:
4^0 ⋅ 5^(-2) / (-2)^(-3) = 1 ⋅ 5^(-2) / (-2)^(-3)
Next, recall that a negative exponent can be converted to a positive exponent by taking the reciprocal of the base. Applying this, we get:
1 ⋅ 1/(5^2) / 1/((-2)^3)
Simplifying further, we have:
1 / (25) / 1 / (-8)
Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore:
1 / (25) / 1 / (-8) = 1 / (25) * (-8)
Finally, multiplying the fractions, we get:
1 * (-8) / 25 = -8 / 25
So the simplified fraction, with no remaining exponents, is -8/25.
First, recall that any number (except 0) raised to the power of 0 is equal to 1. Therefore, we have:
4^0 ⋅ 5^(-2) / (-2)^(-3) = 1 ⋅ 5^(-2) / (-2)^(-3)
Next, recall that a negative exponent can be converted to a positive exponent by taking the reciprocal of the base. Applying this, we get:
1 ⋅ 1/(5^2) / 1/((-2)^3)
Simplifying further, we have:
1 / (25) / 1 / (-8)
Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore:
1 / (25) / 1 / (-8) = 1 / (25) * (-8)
Finally, multiplying the fractions, we get:
1 * (-8) / 25 = -8 / 25
So the simplified fraction, with no remaining exponents, is -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.(1 point)
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents. Let's simplify the expression step by step:
First, recall that any number (except 0) raised to the power of 0 is equal to 1. Therefore, we have:
(14^3 ⋅ 27^(-7))^0 / 3^4 = 1 / 3^4
Next, let's simplify 3^4:
1 / 81
Therefore, the simplified fraction with no remaining exponents is 1/81.
First, recall that any number (except 0) raised to the power of 0 is equal to 1. Therefore, we have:
(14^3 ⋅ 27^(-7))^0 / 3^4 = 1 / 3^4
Next, let's simplify 3^4:
1 / 81
Therefore, the simplified fraction with no remaining exponents is 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.(1 point)
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents. Let's simplify the expression step by step:
First, recall that for any number a, (a^m)^n = a^(m*n). Therefore, we have:
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
Using this property, we can simplify the expression as:
4^(3*(-5)) ⋅ 7^((-2)*(-5)) / 7^9 ⋅ 4^(-13)
Simplifying the exponents, we get:
4^(-15) ⋅ 7^(10) / 7^9 ⋅ 4^(-13)
Next, recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the expression as:
4^(-15) ⋅ 7^(10) ⋅ 7^(-9) ⋅ 4^(-13)
Now, let's simplify the expression further. Using the property a^m ⋅ a^n = a^(m+n), we can combine the terms with the same base:
(4^(-15) ⋅ 4^(-13)) ⋅ (7^(10) ⋅ 7^(-9))
Simplifying the exponents again, we obtain:
4^(-28) ⋅ 7^(1)
Finally, we can rewrite the expression in terms of positive exponents:
(1/4^28) ⋅ 7^1
Simplifying the expression further, we have:
7 / 4^28
Therefore, the simplified fraction with no remaining exponents is 7/4^28.
First, recall that for any number a, (a^m)^n = a^(m*n). Therefore, we have:
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
Using this property, we can simplify the expression as:
4^(3*(-5)) ⋅ 7^((-2)*(-5)) / 7^9 ⋅ 4^(-13)
Simplifying the exponents, we get:
4^(-15) ⋅ 7^(10) / 7^9 ⋅ 4^(-13)
Next, recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the expression as:
4^(-15) ⋅ 7^(10) ⋅ 7^(-9) ⋅ 4^(-13)
Now, let's simplify the expression further. Using the property a^m ⋅ a^n = a^(m+n), we can combine the terms with the same base:
(4^(-15) ⋅ 4^(-13)) ⋅ (7^(10) ⋅ 7^(-9))
Simplifying the exponents again, we obtain:
4^(-28) ⋅ 7^(1)
Finally, we can rewrite the expression in terms of positive exponents:
(1/4^28) ⋅ 7^1
Simplifying the expression further, we have:
7 / 4^28
Therefore, the simplified fraction with no remaining exponents is 7/4^28.
Okay can you write that differently
Certainly! Rewriting the expression using positive exponents, we have:
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
= (4^(-15) ⋅ 7^(10)) / (7^9 ⋅ 4^(-28))
= (7/4^28) ⋅ (4^(-15) ⋅ 7^(10))
= 7/4^28 ⋅ 4^(-15) ⋅ 7^10
= 7/4^28 ⋅ 4^(-15) ⋅ 7^10
Keep in mind that 4^(-13) has been simplified to 4^(-28) in the process.
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
= (4^(-15) ⋅ 7^(10)) / (7^9 ⋅ 4^(-28))
= (7/4^28) ⋅ (4^(-15) ⋅ 7^(10))
= 7/4^28 ⋅ 4^(-15) ⋅ 7^10
= 7/4^28 ⋅ 4^(-15) ⋅ 7^10
Keep in mind that 4^(-13) has been simplified to 4^(-28) in the process.
Sorry Bot but I still don't get it
Apologies for any confusion. Let's simplify the expression step by step:
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
First, we can apply the property (a^m)^n = a^(m*n) to the numerator:
(4^(3*(-5)) ⋅ 7^((-2)*(-5))) / 7^9 ⋅ 4^(-13)
Simplifying the exponents, we get:
(4^(-15) ⋅ 7^(10)) / 7^9 ⋅ 4^(-13)
Next, we can rewrite the expression using positive exponents by taking the reciprocal of the bases with negative exponents:
(1/4^15 ⋅ 7^10) / 7^9 ⋅ 1/4^13
Simplifying further, we have:
(1/4^15 ⋅ 7^10) / (1/7^9 ⋅ 4^13)
Now, dividing by a fraction is equivalent to multiplying by its reciprocal:
(1/4^15 ⋅ 7^10) ⋅ (7^9 ⋅ 4^13)
Simplifying the expression, we get:
(7^10 ⋅ 7^9) ⋅ (4^13 / 4^15)
Combining the terms with the same base, we have:
7^(10+9) ⋅ 4^(13-15)
Simplifying further, we get:
7^19 ⋅ 4^(-2)
Finally, rewriting the expression with positive exponents:
7^19 / 4^2
Therefore, the simplified fraction with no remaining exponents is 7^19 / 4^2.
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
First, we can apply the property (a^m)^n = a^(m*n) to the numerator:
(4^(3*(-5)) ⋅ 7^((-2)*(-5))) / 7^9 ⋅ 4^(-13)
Simplifying the exponents, we get:
(4^(-15) ⋅ 7^(10)) / 7^9 ⋅ 4^(-13)
Next, we can rewrite the expression using positive exponents by taking the reciprocal of the bases with negative exponents:
(1/4^15 ⋅ 7^10) / 7^9 ⋅ 1/4^13
Simplifying further, we have:
(1/4^15 ⋅ 7^10) / (1/7^9 ⋅ 4^13)
Now, dividing by a fraction is equivalent to multiplying by its reciprocal:
(1/4^15 ⋅ 7^10) ⋅ (7^9 ⋅ 4^13)
Simplifying the expression, we get:
(7^10 ⋅ 7^9) ⋅ (4^13 / 4^15)
Combining the terms with the same base, we have:
7^(10+9) ⋅ 4^(13-15)
Simplifying further, we get:
7^19 ⋅ 4^(-2)
Finally, rewriting the expression with positive exponents:
7^19 / 4^2
Therefore, the simplified fraction with no remaining exponents is 7^19 / 4^2.
That still isn't the correct answer for me?
Apologies for any confusion. Let's simplify the expression step by step again:
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
First, we can simplify the exponent inside the parentheses using the property of integer exponents (a^m ⋅ a^n = a^(m+n)):
(4^(3+(-2)))^(-5) / 7^9 ⋅ 4^(-13)
Simplifying the exponent inside the parentheses, we get:
(4^1)^(-5) / 7^9 ⋅ 4^(-13)
Next, any number raised to the power of 1 is equal to itself, so we can simplify further:
4^(-5) / 7^9 ⋅ 4^(-13)
Now, let's simplify the expression with negative exponents. We know that a^(-n) is equal to 1/a^n:
1/4^5 / 7^9 ⋅ 1/4^13
Simplifying each term separately, we get:
1/(1024) / 7^9 ⋅ 1/(16384)
Now, let's multiply the fractions:
1 / (1024 ⋅ 7^9 ⋅ 16384)
Simplifying the expression further, we have:
1 / (1125899906842624 ⋅ 7^9)
Therefore, the simplified fraction with no remaining exponents is 1 / (1125899906842624 ⋅ 7^9).
(4^3 ⋅ 7^(-2))^(-5) / 7^9 ⋅ 4^(-13)
First, we can simplify the exponent inside the parentheses using the property of integer exponents (a^m ⋅ a^n = a^(m+n)):
(4^(3+(-2)))^(-5) / 7^9 ⋅ 4^(-13)
Simplifying the exponent inside the parentheses, we get:
(4^1)^(-5) / 7^9 ⋅ 4^(-13)
Next, any number raised to the power of 1 is equal to itself, so we can simplify further:
4^(-5) / 7^9 ⋅ 4^(-13)
Now, let's simplify the expression with negative exponents. We know that a^(-n) is equal to 1/a^n:
1/4^5 / 7^9 ⋅ 1/4^13
Simplifying each term separately, we get:
1/(1024) / 7^9 ⋅ 1/(16384)
Now, let's multiply the fractions:
1 / (1024 ⋅ 7^9 ⋅ 16384)
Simplifying the expression further, we have:
1 / (1125899906842624 ⋅ 7^9)
Therefore, the simplified fraction with no remaining exponents is 1 / (1125899906842624 ⋅ 7^9).
Nope Bot still not the answer.
I figured it out though it's 7/16
Thanks for trying on that one
I figured it out though it's 7/16
Thanks for trying on that one
Apologies for the confusion earlier and glad to hear you figured out the correct answer. If you have any more questions or need further assistance, feel free to ask.
Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/ 3^−9?(1 point)
Responses
only the Power Rule of Exponents
only the Property of Negative Exponents
the Property of Negative Exponents and the Product Rule of Exponents
only the Product Rule of Exponents
Responses
only the Power Rule of Exponents
only the Property of Negative Exponents
the Property of Negative Exponents and the Product Rule of Exponents
only the Product Rule of Exponents
The correct answer is "the Property of Negative Exponents and the Product Rule of Exponents."
In the given expression 3^5 / 3^-9, the property of negative exponents allows us to rewrite 3^-9 as 1 / 3^9. Then, using the product rule of exponents, we can simplify the expression as 3^5 * (1 / 3^9), which further simplifies to 3^5 / 3^9. Finally, this can be rewritten as 3^(5-9) = 3^(-4) = 1 / 3^4 = 1 / 81.
So, the correct answer is the Property of Negative Exponents and the Product Rule of Exponents.
In the given expression 3^5 / 3^-9, the property of negative exponents allows us to rewrite 3^-9 as 1 / 3^9. Then, using the product rule of exponents, we can simplify the expression as 3^5 * (1 / 3^9), which further simplifies to 3^5 / 3^9. Finally, this can be rewritten as 3^(5-9) = 3^(-4) = 1 / 3^4 = 1 / 81.
So, the correct answer is the Property of Negative Exponents and the Product Rule of 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
(−4)^3 / 7^2
7^2 / (−4)^3
4^3/ 7^2
1/ 7^2 ⋅ (−4)^−3
Responses
(−4)^3 / 7^2
7^2 / (−4)^3
4^3/ 7^2
1/ 7^2 ⋅ (−4)^−3
The correct answer is "7^2 / (−4)^3".
To generate an equivalent expression with only positive exponents, we can use the property of integer exponents that states:
a^m / a^n = a^(m - n)
Using this property, we can rewrite the expression 15^0 ⋅ 7^(-2) / (-4)^(-3) as:
1 / 7^2 ⋅ (-4)^3
Now, we can rewrite (-4)^3 as (-4)^3 = -(4^3) = -(64) = -64.
Therefore, the equivalent expression with only positive exponents is 7^2 / (-4)^3.
To generate an equivalent expression with only positive exponents, we can use the property of integer exponents that states:
a^m / a^n = a^(m - n)
Using this property, we can rewrite the expression 15^0 ⋅ 7^(-2) / (-4)^(-3) as:
1 / 7^2 ⋅ (-4)^3
Now, we can rewrite (-4)^3 as (-4)^3 = -(4^3) = -(64) = -64.
Therefore, the equivalent expression with only positive exponents is 7^2 / (-4)^3.
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^−7⋅ 9^−1 / 2^9⋅9^−10
2^12⋅ 9^−12 / 2^9⋅ 9^−10
2^3 ⋅9^2
2^3 / 9^2
Responses
2^−7⋅ 9^−1 / 2^9⋅9^−10
2^12⋅ 9^−12 / 2^9⋅ 9^−10
2^3 ⋅9^2
2^3 / 9^2
The correct answer is "2^(-7)⋅9^(-1) / 2^9⋅9^(-10)".
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents. Let's simplify the given expression step by step:
First, we can simplify the exponent inside the parentheses using the property of integer exponents: (a^m)^n = a^(m*n).
(2^(-3)⋅9^3)^(-4) / 2^9⋅9^(-10) rewrittes as: 2^((-3)(-4))⋅9^(3*(-4)) / 2^9⋅9^(-10).
Simplifying the exponents, we have: 2^12⋅9^(-12) / 2^9⋅9^(-10).
Next, recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Using this, we can rewrite the expression as: 2^12⋅9^(-12) ⋅ (2^9⋅9^(-10))^(-1).
Now, we can use the property a^(-n) = 1/a^n to rewrite (2^9⋅9^(-10))^(-1) as 1/(2^9⋅9^(-10)).
Combining the terms and simplifying further, we get: 2^12⋅9^(-12) / (2^9⋅9^(-10)).
To simplify this expression with positive exponents, we can change the base with negative exponents to its reciprocal. Applying this, we have: 2^12⋅(1/9^12) / (2^9⋅(1/9^10)).
Simplifying further, we obtain: 2^12/9^12 ⋅ 9^10/2^9.
Finally, rewriting the expression with positive exponents, we get: 2^3⋅9^10 / 9^12⋅2^(-9).
Therefore, the equivalent expression with only positive exponents is 2^3⋅9^10 / 9^12⋅2^(-9), which matches the option 2^3 / 9^2.
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents. Let's simplify the given expression step by step:
First, we can simplify the exponent inside the parentheses using the property of integer exponents: (a^m)^n = a^(m*n).
(2^(-3)⋅9^3)^(-4) / 2^9⋅9^(-10) rewrittes as: 2^((-3)(-4))⋅9^(3*(-4)) / 2^9⋅9^(-10).
Simplifying the exponents, we have: 2^12⋅9^(-12) / 2^9⋅9^(-10).
Next, recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Using this, we can rewrite the expression as: 2^12⋅9^(-12) ⋅ (2^9⋅9^(-10))^(-1).
Now, we can use the property a^(-n) = 1/a^n to rewrite (2^9⋅9^(-10))^(-1) as 1/(2^9⋅9^(-10)).
Combining the terms and simplifying further, we get: 2^12⋅9^(-12) / (2^9⋅9^(-10)).
To simplify this expression with positive exponents, we can change the base with negative exponents to its reciprocal. Applying this, we have: 2^12⋅(1/9^12) / (2^9⋅(1/9^10)).
Simplifying further, we obtain: 2^12/9^12 ⋅ 9^10/2^9.
Finally, rewriting the expression with positive exponents, we get: 2^3⋅9^10 / 9^12⋅2^(-9).
Therefore, the equivalent expression with only positive exponents is 2^3⋅9^10 / 9^12⋅2^(-9), which matches the option 2^3 / 9^2.