Squaring a number and taking the square root of a number are inverse operations. When taking a square root, the number of solutions can be different.
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Practice makes perfect
We want to determine the number of solutions of the given equation without solving it.
b^2=- sqrt(1/9)
First, let's recall some important facts relating to solving equations involving square roots. Consider an example equation.
a^2=49
In this expression, the a-term is squared, so we need to take the square root of both sides. When doing so, keep in mind that we take both the positive and the negative square roots. This means that equations that look like this have two solutions, one positive and one negative. Next, let's move to another example.
b=- sqrt(16)
This time, the expression on the right-hand side of the equation represents the negative square root. With different representations of the square root symbol, we can expect a different number of solutions. Let's take a look at a table that shows this.
Representation
Square Root
Example
sqrt()
Positive
sqrt(4)=2
- sqrt()
Negative
- sqrt(4)=-2
± sqrt()
Both
± sqrt(4)=2 and -2
As we can see, there can be one or two solutions when taking a square root. Remember that the square root of a negative number is not a real number because there is no real number that results in a negative number when multiplied by itself. Now, with all of this in mind, let's return to the given equation.
b^2=-sqrt(1/9)
To solve this equation, we need to take the square root of both sides because squaring a number and taking the square root of a number are inverse operations. Notice that the expression on the right-hand side is a negative number. We noted before that the square root of a negative number is not a real number. That means there are 0 solutions to this equation.
