Solve the following equation for x: 4^x - 16^(x-2)=0

To solve this equation, we notice that 16 is 4^2. So, we change the 16 to 4^2 which leads to the following equation: 4^x - 4^(2x-4)=0 move one of the 4s to the other side. It becomes: 4^(x) = 4^(2x-4) we apply log (of base 4) on both sides to get rid of the "4" because log of a number to the same number is 1. Therefore, x=2x-4, solving for x leads to x=4

A right-angle triangle has a base of 4 cm and a hypotenuse of 5 cm. What is its height?

Well, since this is a right-angle triangle, we can apply Pythagorean Theorem, which states that the (Hypotenuse)^2= (base)^2+(height)^2 base =4 cm Hypotenuse = 5 cm. Therefore, 5^2=4^2+height^2 (height)^2= 25 - 16= 9 Therefore, the Height is= sqrt (9)= 3 cm

A cube has a volume of 125 cm^3. If its width is 3x^2-22, its length is -x+8, and its height is x^2-4. Find x?

Since the object in question is a cube, that means each of its sides must be the same length. Therefore, to get a volume of 125 cm^3, each side must be equal to the cube root of 125 cm^3, which is 5 cm. So, we can set each side to 5 cm. It becomes 3x^2-22=5, solving for x, we get x=3 cm. To verify, the length is -x+7 by substituting x=3, we get the length of -3+8= 5 cm. Similarly, the height is (3)^2-4=9-4=5 cm