x² = 49 and x³ = 27
Today we'll solve equations like x squared equals forty-nine. What number, multiplied by itself, gives forty-nine?
We write the equation: x squared equals forty-nine. Our job is to find x.
Think back: seven times seven is forty-nine. So seven works!
To undo the square, we take the square root of both sides.
That gives us x equals seven.
But wait — negative seven times negative seven is also forty-nine!
So x squared equals forty-nine has two solutions: positive seven AND negative seven. We write x equals plus or minus seven.
Cubes work differently. Let's try x cubed equals twenty-seven.
Three times three times three is twenty-seven, so x equals three.
Could x be negative three? No — negative three cubed is negative twenty-seven, not twenty-seven. Cube roots give just one answer.
Square roots have two solutions, plus and minus. Cube roots have one. Great work!