The square root of 169 is 13. A square root symbol typically refers to the principal (non-negative) square root. Thus, the correct response is:
13
However, if the question allowed for both the positive and negative square roots, then the correct responses would be:
13, −13
The response 84.5 is incorrect for the square root of 169.
Find the square root of 169: √169
(1 point)
Responses
13
13
−13
negative 13
13,−13
13 comma negative 13
84.5
11 answers
Find the cube root of −729: ^3√−729
Responses
9
9
−9
negative 9
243
243
9, −9
Responses
9
9
−9
negative 9
243
243
9, −9
The cube root of a negative number is also negative because a negative number raised to the power of 3 gives a negative result. Therefore, the cube root of −729 is:
−9
The correct response is:
−9
negative 9
The response "9, −9" is incorrect because cube roots yield a single value for each real number. The response "243" is incorrect as well; 243 is 3 raised to the power of 5, not the cube root of any number. The response "9" is incorrect because it is the positive cube root of +729, not −729.
−9
The correct response is:
−9
negative 9
The response "9, −9" is incorrect because cube roots yield a single value for each real number. The response "243" is incorrect as well; 243 is 3 raised to the power of 5, not the cube root of any number. The response "9" is incorrect because it is the positive cube root of +729, not −729.
Find the cube root of −8 : ^3√-8(1 point)
The cube root of -8 is:
−2
This is because (-2) × (-2) × (-2) equals -8. Each factor (-2) is multiplied by itself three times to give the original number, -8.
So the correct response is:
−2
negative 2
−2
This is because (-2) × (-2) × (-2) equals -8. Each factor (-2) is multiplied by itself three times to give the original number, -8.
So the correct response is:
−2
negative 2
Find the cubic root of 135.(1 point)
Responses
35–√
3 start root 5 end root
527−−√3
5 root index 3 start root 27 end root
5–√3
root index 3 start root 5 end root
35–√3
Responses
35–√
3 start root 5 end root
527−−√3
5 root index 3 start root 27 end root
5–√3
root index 3 start root 5 end root
35–√3
To find the cube root of 135, we can try to simplify it by finding the largest perfect cube that divides 135. The largest perfect cube in this case is 27 (since 3^3 = 27), which is a factor of 135. We can express 135 as the product of 27 and 5:
135 = 27 × 5
Now, we can take the cube root of both factors:
^3√135 = ^3√(27 × 5)
Since the cube root of 27 is 3, and the cube root of 5 cannot be simplified further, the result is:
^3√135 = 3 × ^3√5
None of the responses provided in the question are formatted correctly for standard interpretation, but if we attempt to match the correct cube root to the given options, "5–√3" seems to be an attempt to represent 5 times the cube root of 3, which is not correct for the cube root of 135. The correct cubic root representation for 135 should be "3√5".
So the response that most accurately represents the cube root of 135 is:
3√5
If none of the given options are correct and a point must be awarded, please provide the standard notation or the context in which these responses are given for further clarification.
135 = 27 × 5
Now, we can take the cube root of both factors:
^3√135 = ^3√(27 × 5)
Since the cube root of 27 is 3, and the cube root of 5 cannot be simplified further, the result is:
^3√135 = 3 × ^3√5
None of the responses provided in the question are formatted correctly for standard interpretation, but if we attempt to match the correct cube root to the given options, "5–√3" seems to be an attempt to represent 5 times the cube root of 3, which is not correct for the cube root of 135. The correct cubic root representation for 135 should be "3√5".
So the response that most accurately represents the cube root of 135 is:
3√5
If none of the given options are correct and a point must be awarded, please provide the standard notation or the context in which these responses are given for further clarification.
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
.(1 point)
Responses
14.2 miles
14.2 miles
63.59 miles
63.59 miles
14.13 miles
14.13 miles
7.07 miles
.(1 point)
Responses
14.2 miles
14.2 miles
63.59 miles
63.59 miles
14.13 miles
14.13 miles
7.07 miles
To find the circumference of a circle, you can use the formula:
Circumference = π × Diameter
Given that the diameter of the circle is 4.5 miles and you're asked to use 3.14 for π, the calculation would be:
Circumference = 3.14 × 4.5 miles
Now, let's do the multiplication:
Circumference ≈ 14.13 miles
When rounding to the hundredths place, 14.13 is already at that precision, so there's no further rounding needed.
The correct estimate to the hundredths place for the circumference is:
14.13 miles
Circumference = π × Diameter
Given that the diameter of the circle is 4.5 miles and you're asked to use 3.14 for π, the calculation would be:
Circumference = 3.14 × 4.5 miles
Now, let's do the multiplication:
Circumference ≈ 14.13 miles
When rounding to the hundredths place, 14.13 is already at that precision, so there's no further rounding needed.
The correct estimate to the hundredths place for the circumference is:
14.13 miles
Rewrite the irrational cube root 189−−−√3 as the product of an integer and another irrational cube root. Show your work.
Note: You will find the option to add a cubic root symbol in the Algebra ( ×
) keyboard.
(2 points)
Note: You will find the option to add a cubic root symbol in the Algebra ( ×
) keyboard.
(2 points)
To rewrite the cube root of 189 as the product of an integer and another irrational cube root, we need to factor 189 into a product of a perfect cube and another number. Let's find the prime factorization of 189 first:
189 = 3 × 63
189 = 3 × 3 × 21
189 = 3 × 3 × 3 × 7
189 = 3^3 × 7
Now that we have the prime factorization, we see that 189 contains the perfect cube 3^3, which is 27. Let's express 189 as the product of this perfect cube and 7:
189 = 27 × 7
We can now rewrite the cube root of 189:
^3√189 = ^3√(27 × 7)
Since the cube root of 27 is 3, we separate the perfect cube from the number that remains under the cube root:
^3√189 = ^3√(3^3) × ^3√7
^3√189 = 3 × ^3√7
Thus, the irrational cube root ^3√189 can be rewritten as the product of the integer 3 and the irrational cube root of 7:
3 × ^3√7
189 = 3 × 63
189 = 3 × 3 × 21
189 = 3 × 3 × 3 × 7
189 = 3^3 × 7
Now that we have the prime factorization, we see that 189 contains the perfect cube 3^3, which is 27. Let's express 189 as the product of this perfect cube and 7:
189 = 27 × 7
We can now rewrite the cube root of 189:
^3√189 = ^3√(27 × 7)
Since the cube root of 27 is 3, we separate the perfect cube from the number that remains under the cube root:
^3√189 = ^3√(3^3) × ^3√7
^3√189 = 3 × ^3√7
Thus, the irrational cube root ^3√189 can be rewritten as the product of the integer 3 and the irrational cube root of 7:
3 × ^3√7
4 answers