What is the principle of powers? Describe in your own words. When solving a radical equation, how do we use the principle of powers to eliminate radicals? Demonstrate with an example.
roots and powers are inverse operations, just like add/subtract and multiply/divide
2+6=8 can be written 2=8-6 or 6=8-2
2*6=12 can be written as 12/6-2 or 12/2=6
sqrt(9) = 9^(1/2)
[sqrt(9)]^2 = [9^(1/2)]^2
9 = 9^[1/2 * 2] = 9^1 = 9
Roots are fractional powers
x^(m/n) is the nth root of x^m
or, it's [x^(1/n)]^m
So, whenever you have a radical involved, to get rid of it, raise everything to the power of the index
If you have a power of n, take the nth root of everything to get rid of the exponent.
It's just like having 6x+3=9
to get rid of the 3, subtract from both sides; then to get rid of the 6, divide both sides by 6.
Powers and roots work just the same way.
The principle of powers refers to the property of exponents that allows us to simplify expressions or equations involving radicals (also known as square roots). It states that if an equation contains a radical expression, we can raise both sides of the equation to a power that is the reciprocal of the radical's index to eliminate the radical.
To solve a radical equation using the principle of powers, we follow these steps:
1. Identify the radical expression in the equation.
2. Determine the index of the radical, which tells us the power to which we need to raise both sides of the equation.
3. Raise both sides of the equation to the power that is the reciprocal of the radical's index.
4. Simplify the equation by applying the power to each term.
5. Solve the resulting equation for the variable.
6. Check the solution(s) in the original equation to verify their validity.
Let's demonstrate this process with an example:
Example: Solve the equation √(2x + 4) = 6
1. Identify the radical expression in the equation: √(2x + 4)
2. Determine the index of the radical: The index for a square root is 2, so we need to raise both sides to the power of 2.
3. Raise both sides to the power of 2: (√(2x + 4))^2 = 6^2, simplify to 2x + 4 = 36.
4. Simplify the equation: 2x + 4 = 36.
5. Solve the resulting equation for the variable: Subtract 4 from both sides: 2x = 36 - 4 = 32. Divide both sides by 2: x = 32/2 = 16.
6. Check the solution: Substitute x = 16 into the original equation: √(2(16) + 4) = 6. Simplify: √(32 + 4) = 6, √36 = 6, 6 = 6. The solution x = 16 is valid.
Therefore, the solution to the equation √(2x + 4) = 6 is x = 16.