What would the ⍰ need to be in the expression below for the simplified form of the expression to be equal to
1/8xy^6?
4x^2y^-5/2⍰x^3y
The ⍰ would need to be equal to y^12 in order for the simplified form of the expression to be equal to 1/8xy^6.
Why does the equation 3x^2 - 30 = 3 have two solutions but the equation 3x^2 + 30 = 3 has no solution?
Try some different values for x.
Let's solve the equations to determine why the first equation has two solutions and the second equation has no solution.
1) 3x^2 - 30 = 3
First, we'll add 30 to both sides to isolate the x^2 term:
3x^2 = 33
Then, divide both sides by 3 to solve for x^2:
x^2 = 11
Finally, take the square root of both sides to find the two possible solutions for x:
x = ±√(11)
Thus, the equation 3x^2 - 30 = 3 has two solutions: x = √(11) and x = -√(11).
2) 3x^2 + 30 = 3
Similar to the previous equation, we'll subtract 30 from both sides to isolate the x^2 term:
3x^2 = -27
Next, divide both sides by 3:
x^2 = -9
Now, when we take the square root of both sides, we encounter an issue. The square root of a negative number (√(-9)) does not yield a real number. Therefore, there are no real solutions for x in the equation 3x^2 + 30 = 3.
Trying different values for x won't change the fact that the equation has no real solutions.
In the expression below if a = 4, what value of b is needed for the expression to have a value of 81?
12a^3b^7(ab)^-4
To find the value of b that makes the expression equal to 81, we can substitute a=4 into the expression and solve for b.
Given: 12a^3b^7(ab)^-4 = 81
Substituting a=4:
12(4)^3b^7(4b)^-4 = 81
Simplifying:
12(64)b^7(4b)^-4 = 81
768b^7(4b)^-4 = 81
Now, let's simplify the expression inside the brackets (4b)^-4:
(4b)^-4 = (1/(4b)^4) = (1/256b^4)
Substituting back into the equation:
768b^7 * (1/256b^4) = 81
Simplifying further:
3b^3 = 81
Dividing both sides by 3:
b^3 = 27
Taking the cube root of both sides:
b = 3
Therefore, when a = 4, the value of b needed for the expression to have a value of 81 is 3.
A cube has side lengths of 2^5 cm long. If Sam has 16 of these cubes, what total volume would the cubes occupy written as a power of 2?
The total volume is 2____ cm3
The formula to calculate the volume of a cube is V = s^3, where s is the length of the side.
In this case, each cube has a side length of 2^5 cm, so the volume of one cube is (2^5)^3 = 2^15 cm^3.
Since Sam has 16 cubes, the total volume would be 16 times the volume of one cube:
Total volume = 16 * (2^15 cm^3) = 2^(4 + 15) cm^3 = 2^19 cm^3.
Therefore, the total volume of the cubes would occupy 2^19 cm^3.
Kelly counted all the doors and wheels on her property. She counted 8^4 doors and 8^2 wheels. How many times more doors are there than wheels?
To find out how many times more doors there are than wheels, we need to divide the number of doors by the number of wheels.
Given that Kelly counted 8^4 doors and 8^2 wheels, we have:
Number of doors = 8^4
Number of wheels = 8^2
To calculate the ratio of doors to wheels, we divide the number of doors by the number of wheels:
Ratio = (Number of doors) / (Number of wheels)
Ratio = (8^4) / (8^2)
When dividing exponents with the same base, we subtract the exponents:
Ratio = 8^(4 - 2) = 8^2
Therefore, there are 8^2 times more doors than wheels.