How do you solve for x in this radical equation?

  • Thread starter Quasaire
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In summary, the process for solving a radical equation involves isolating the radical term, eliminating it by raising both sides of the equation to a power, and solving for x. There may be restrictions to keep in mind, such as avoiding negative values under the radical or division by zero. If there are multiple radicals, they can be solved one at a time by simplifying and checking for restrictions.
  • #1
Quasaire
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sqrt[x]{64} = 4

How do you solve for x?

I mean obviously the answer is x = 3 but how do you prove this algebraically?
 
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  • #2
You can use logarithms or 'play' with powers:

[tex]\sqrt[x]{{64}} = 4 \Leftrightarrow 64^{1/x} = 4 \Leftrightarrow \left( {4^3 } \right)^{1/x} = 4 \Leftrightarrow 4^{3/x} = 4^1 \Leftrightarrow \frac{3}
{x} = 1[/tex]
 
  • #3
I really, really wish people would say "x-root" rather than "x-square root" or root[x} instead of (as here) sqrt[x]. "square root" means specifically
[tex]^2\sqrt{x}[/tex]
 

1. How do you isolate the radical term?

The first step in solving a radical equation is to isolate the radical term on one side of the equation. To do this, you may need to combine like terms on the other side of the equation or move them to the other side. Once the radical term is isolated, you can proceed with solving for x.

2. What is the process for eliminating the radical?

To eliminate the radical, you need to raise both sides of the equation to a power that will cancel out the radical. For example, if the radical is a square root, you would raise both sides to the power of 2. This will result in the radical term disappearing on one side of the equation, leaving you with a simpler equation to solve for x.

3. Can you give an example of solving a radical equation?

Sure! Let's say we have the equation √(x+3) = 5. To isolate the radical term, we would subtract 3 from both sides, giving us √x = 2. Then, we can eliminate the radical by squaring both sides, resulting in x = 4.

4. Are there any restrictions when solving radical equations?

Yes, there are some restrictions to keep in mind. When solving radical equations, you must ensure that the value inside the radical is not negative, as this would result in an imaginary solution. Additionally, if the equation includes a variable in the denominator of the radical, you must make sure to exclude any values that would result in division by zero.

5. What should I do if the equation has multiple radicals?

If the equation has more than one radical, you can follow the same steps as before, isolating each radical term one at a time and eliminating them. It may also be helpful to simplify the radicals before solving, if possible. Remember to check for any restrictions on each radical before finalizing your solution.

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