Question
11√19 - 5√19
whats the difference?
whats the difference?
Answers
GPT 3.5
The difference between 11√19 and 5√19 is 6√19.
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multiply (5√19) (6√2) give an exact answer
GPT 3.5
To multiply (5√19) and (6√2), we can apply the distributive property.
(5√19) (6√2) = 5*6*(√19*√2) = 30√(19*2) = 30√38
Therefore, the exact answer is 30√38.
(5√19) (6√2) = 5*6*(√19*√2) = 30√(19*2) = 30√38
Therefore, the exact answer is 30√38.
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simplify √48
GPT 3.5
To simplify √48, we need to find the largest perfect square that is a factor of 48. We can see that 16 is the largest perfect square that divides 48.
√48 = √(16 * 3)
Since the square root can be distributed over multiplication, we can simplify further:
√(16 * 3) = √16 * √3 = 4√3
Therefore, √48 simplifies to 4√3.
√48 = √(16 * 3)
Since the square root can be distributed over multiplication, we can simplify further:
√(16 * 3) = √16 * √3 = 4√3
Therefore, √48 simplifies to 4√3.
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simplify 3^√54
GPT 3.5
To simplify 3^√54, we can rewrite 54 as a product of perfect square factors:
54 = 9 * 6 = 3^2 * 2 * 3
Now we can rewrite 3^√54 using exponent rules:
3^√54 = 3^(√(3^2 * 2 * 3))
Since the square root (denoted by √) and the exponent of 2 are inverse operations, we can simplify further:
3^(√(3^2 * 2 * 3)) = 3^(√(3^2) * √(2) * √(3))
Since √(3^2) is simply 3 and √(2) and √(3) cannot be simplified further, we have:
3^(√(3^2) * √(2) * √(3)) = 3^(3 * √(2) * √(3))
Therefore, the simplified form of 3^√54 is 3^(3 * √(2) * √(3)).
54 = 9 * 6 = 3^2 * 2 * 3
Now we can rewrite 3^√54 using exponent rules:
3^√54 = 3^(√(3^2 * 2 * 3))
Since the square root (denoted by √) and the exponent of 2 are inverse operations, we can simplify further:
3^(√(3^2 * 2 * 3)) = 3^(√(3^2) * √(2) * √(3))
Since √(3^2) is simply 3 and √(2) and √(3) cannot be simplified further, we have:
3^(√(3^2) * √(2) * √(3)) = 3^(3 * √(2) * √(3))
Therefore, the simplified form of 3^√54 is 3^(3 * √(2) * √(3)).
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find the square root of 169: √169
GPT 3.5
The square root of 169 is 13.
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what is the irrational square root √63 rewritten as the product of integer and another irrational square root, in simplest form
GPT 3.5
To rewrite √63 as the product of an integer and another irrational square root in simplest form, we need to find the largest perfect square that divides 63.
By observing, we can see that 9 is the largest perfect square that divides 63.
So, we can rewrite √63 as √(9 * 7).
Now, using the property of the square root, we can split it into two separate square roots:
√(9 * 7) = √9 * √7 = 3√7.
Therefore, √63 can be rewritten as the product of 3 and √7, in simplest form.
By observing, we can see that 9 is the largest perfect square that divides 63.
So, we can rewrite √63 as √(9 * 7).
Now, using the property of the square root, we can split it into two separate square roots:
√(9 * 7) = √9 * √7 = 3√7.
Therefore, √63 can be rewritten as the product of 3 and √7, in simplest form.
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find the cubic root of 135
GPT 3.5
The cubic root of 135 is approximately 5.260.
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give exact answer
GPT 3.5
To find the exact cubic root of 135, we need to determine the prime factorization of 135.
135 = 3 * 3 * 3 * 5
Now, we can group the factors in sets of three to find the cube root.
∛135 = ∛(3 * 3 * 3 * 5) = (∛3 * ∛3 * ∛3) * ∛5 = 3∛5.
Therefore, the exact cubic root of 135 is 3∛5.
135 = 3 * 3 * 3 * 5
Now, we can group the factors in sets of three to find the cube root.
∛135 = ∛(3 * 3 * 3 * 5) = (∛3 * ∛3 * ∛3) * ∛5 = 3∛5.
Therefore, the exact cubic root of 135 is 3∛5.
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divide √95/3√5 provide exact answer
GPT 3.5
To divide √95 by 3√5, we can simplify by multiplying both the numerator and denominator by the conjugate of the denominator, which is √5.
√95/3√5 = (√95/3√5) * (√5/√5)
Simplifying the numerator and denominator:
= (√(95*5))/(3 * √(5*5))
= √475/3√25
= √475/3√(5*5)
= √475/3 * 5
= √475/15
Therefore, the exact answer is √475/15.
√95/3√5 = (√95/3√5) * (√5/√5)
Simplifying the numerator and denominator:
= (√(95*5))/(3 * √(5*5))
= √475/3√25
= √475/3√(5*5)
= √475/3 * 5
= √475/15
Therefore, the exact answer is √475/15.
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divide 38√39/9√13
GPT 3.5
To divide 38√39 by 9√13, we can simplify by multiplying both the numerator and denominator by the conjugate of the denominator, which is √13.
(38√39)/(9√13) = (38√39)/(9√13) * (√13/√13)
Simplifying the numerator and denominator:
= (38√(39*13))/(9 * √(13*13))
= (38√(507))/(9 * 13)
= (38√(507))/(9 * 13)
= (2 * 19 * √507) / (3 * 13)
= (38 * √507)/39
Therefore, the simplified form of 38√39/9√13 is (38 * √507)/39.
(38√39)/(9√13) = (38√39)/(9√13) * (√13/√13)
Simplifying the numerator and denominator:
= (38√(39*13))/(9 * √(13*13))
= (38√(507))/(9 * 13)
= (38√(507))/(9 * 13)
= (2 * 19 * √507) / (3 * 13)
= (38 * √507)/39
Therefore, the simplified form of 38√39/9√13 is (38 * √507)/39.
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find the rubic cube of -729: 3^√-729 give exact answer
GPT 3.5
To find the exact cubic root of -729, let's first determine the prime factorization of -729.
-729 can be written as -1 * 3 * 3 * 3 * 3 * 3 * 3 * 3.
Since we're looking for the cubic root, we can group the factors in sets of three:
∛(-729) = ∛(-1 * 3 * 3 * 3 * 3 * 3 * 3 * 3) = -1 * ∛(3 * 3 * 3 * 3 * 3 * 3 * 3)
We know that the cubic root of a negative number is the negative of the cubic root of the positive number. And since we have an odd number of 3's being cubed, the cube root of 3^7 will be negative:
-1 * ∛(3 * 3 * 3 * 3 * 3 * 3 * 3) = -1 * 3 * 3 = -9.
Therefore, the exact cubic root of -729 is -9.
-729 can be written as -1 * 3 * 3 * 3 * 3 * 3 * 3 * 3.
Since we're looking for the cubic root, we can group the factors in sets of three:
∛(-729) = ∛(-1 * 3 * 3 * 3 * 3 * 3 * 3 * 3) = -1 * ∛(3 * 3 * 3 * 3 * 3 * 3 * 3)
We know that the cubic root of a negative number is the negative of the cubic root of the positive number. And since we have an odd number of 3's being cubed, the cube root of 3^7 will be negative:
-1 * ∛(3 * 3 * 3 * 3 * 3 * 3 * 3) = -1 * 3 * 3 = -9.
Therefore, the exact cubic root of -729 is -9.
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find the cube root of -8: 3^√-8 give exact answer
GPT 3.5
To find the exact cubic root of -8, let's first determine the prime factorization of -8.
-8 can be written as -1 * 2 * 2 * 2.
Since we're looking for the cubic root, we can group the factors in sets of three:
∛(-8) = ∛(-1 * 2 * 2 * 2) = -1 * ∛(2 * 2 * 2)
We know that the cubic root of a negative number is the negative of the cubic root of the positive number. Since we have an even number of 2's being cubed, the cube root of 2^3 will be positive:
-1 * ∛(2 * 2 * 2) = -1 * 2 = -2.
Therefore, the exact cubic root of -8 is -2.
-8 can be written as -1 * 2 * 2 * 2.
Since we're looking for the cubic root, we can group the factors in sets of three:
∛(-8) = ∛(-1 * 2 * 2 * 2) = -1 * ∛(2 * 2 * 2)
We know that the cubic root of a negative number is the negative of the cubic root of the positive number. Since we have an even number of 2's being cubed, the cube root of 2^3 will be positive:
-1 * ∛(2 * 2 * 2) = -1 * 2 = -2.
Therefore, the exact cubic root of -8 is -2.
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what is the irrational square root 3^√81 rewritten as the product of integer and another irrational square root, give exact answer
GPT 3.5
To rewrite √(3^√81) as the product of an integer and another irrational square root in exact form, we need to determine the prime factorization of 81.
81 = 3 * 3 * 3 * 3 = 3^4
Now, we can rewrite √(3^√81) using the property of square roots and exponents:
√(3^√81) = √(3^4)^√1 = √(3^4) = 3^2.
Therefore, √(3^√81) can be rewritten as the product of 3 and √1, which simplifies to 3.
81 = 3 * 3 * 3 * 3 = 3^4
Now, we can rewrite √(3^√81) using the property of square roots and exponents:
√(3^√81) = √(3^4)^√1 = √(3^4) = 3^2.
Therefore, √(3^√81) can be rewritten as the product of 3 and √1, which simplifies to 3.
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What is the circumference of a circle with a diameter of 4.5 miles? Estimate the circumference to the hundredths place using 3.14 for π .
GPT 3.5
The circumference of a circle can be calculated using the formula C = πd, where C is the circumference and d is the diameter.
Given that the diameter is 4.5 miles, we can substitute this value into the formula:
C = 3.14 * 4.5
Calculating this, we have:
C ≈ 14.13
Therefore, the circumference of the circle is approximately 14.13 miles.
Given that the diameter is 4.5 miles, we can substitute this value into the formula:
C = 3.14 * 4.5
Calculating this, we have:
C ≈ 14.13
Therefore, the circumference of the circle is approximately 14.13 miles.
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Rewrite the irrational cube root ∛189 as the product of an integer and another irrational cube root. Show your work.
GPT 3.5
To rewrite ∛189 as the product of an integer and another irrational cube root, we need to determine the prime factorization of 189.
189 can be factored as 3 * 3 * 3 * 7.
Now, let's group the factors in sets of three to find the cube root:
∛189 = ∛(3 * 3 * 3 * 7) = ∛(3^3 * 7).
Using the property of the cube root, we can simplify this further:
∛(3^3 * 7) = ∛(3^3) * ∛7 = 3∛7.
Therefore, ∛189 can be rewritten as the product of 3 and ∛7.
189 can be factored as 3 * 3 * 3 * 7.
Now, let's group the factors in sets of three to find the cube root:
∛189 = ∛(3 * 3 * 3 * 7) = ∛(3^3 * 7).
Using the property of the cube root, we can simplify this further:
∛(3^3 * 7) = ∛(3^3) * ∛7 = 3∛7.
Therefore, ∛189 can be rewritten as the product of 3 and ∛7.