- #1
paulmdrdo1
- 385
- 0
Can you explain the part of the solution where v2/v1 = a2/a1were being squared and cubed repectively?
Thanks!
Squaring and Cubing Equations: Explained!
- MHB
- Thread starter paulmdrdo1
- Start date
In summary, squaring an equation involves multiplying it by itself, while cubing an equation involves multiplying it by itself three times. This results in different powers of the variables in the equation. Understanding squaring and cubing equations is important because it allows for solving more complex equations and analyzing relationships between variables more efficiently. To square an equation, it is multiplied by itself, while cubing an equation involves multiplying it by itself three times. These concepts are used in various fields such as physics, engineering, and economics to model real-life situations and make predictions about future outcomes.
Can you explain the part of the solution where v2/v1 = a2/a1were being squared and cubed repectively?
Thanks!
- #2
mrtwhs
- 47
- 0
\(\displaystyle (r^3)^2 = (r^2)^3\)
- #3
Evgeny.Makarov
Gold Member
MHB
- 2,436
- 4
To provide a little more detail: It is known that the volume of a sphere of radius $r$ is $cr^3$ for some $c$ (in fact, $c=\frac43\pi$) and its area is $dr^2$ ($d=4\pi$). Therefore,
\[
\left(\frac{V_2}{V_1}\right)^2=\left(\frac{cr_2^3}{cr_1^2}\right)^2=\left(\frac{r_2}{r_1}\right)^6
\]
One can similarly show that $(A_2/A_1)^3=(r_2/r_1)^6$ regardless of $d$.
\[
\left(\frac{V_2}{V_1}\right)^2=\left(\frac{cr_2^3}{cr_1^2}\right)^2=\left(\frac{r_2}{r_1}\right)^6
\]
One can similarly show that $(A_2/A_1)^3=(r_2/r_1)^6$ regardless of $d$.
FAQ: Squaring and Cubing Equations: Explained!
What is the difference between squaring and cubing an equation?
Squaring an equation means to multiply it by itself, while cubing an equation means to multiply it by itself three times. This results in different powers of the variables in the equation.
Why is it important to understand squaring and cubing equations?
Understanding squaring and cubing equations is important because it allows us to solve more complex equations and analyze relationships between variables in a more efficient way.
How do you square an equation?
To square an equation, you simply multiply the equation by itself. For example, if the equation is x + 2 = 5, squaring it would result in (x + 2)(x + 2) = 5(5).
Can you provide an example of a cubed equation?
One example of a cubed equation is (x + 3)^3 = 125. This means that x + 3 is cubed, or multiplied by itself three times, resulting in 125.
How are squaring and cubing equations used in real life?
Squaring and cubing equations are used in various fields such as physics, engineering, and economics. They can be used to model real-life situations and make predictions about future outcomes.