Solve the following equation: [(2x^(1/2))) - (x^(1/4)) = 0]

The equation you provided is:

\[
(2x^{\frac{1}{2}}) - (x^{\frac{1}{4}}) = 0
\]

Let's solve it step by step:

### Step 1: Isolate the terms
Move one of the terms to the other side of the equation:

\[
2x^{\frac{1}{2}} = x^{\frac{1}{4}}
\]

### Step 2: Express the exponents as powers
We can rewrite this in a more manageable form by expressing both sides as powers of \(x\):

\[
2(x^{\frac{1}{2}}) = x^{\frac{1}{4}}
\]

### Step 3: Divide both sides by \(x^{\frac{1}{4}}\)
Divide both sides by \(x^{\frac{1}{4}}\) to simplify the equation:

\[
2x^{\frac{1}{2} - \frac{1}{4}} = 1
\]

Simplify the exponent:

\[
2x^{\frac{1}{4}} = 1
\]

### Step 4: Solve for \(x^{\frac{1}{4}}\)
Now, divide both sides by 2:

\[
x^{\frac{1}{4}} = \frac{1}{2}
\]

### Step 5: Raise both sides to the power of 4
To get \(x\) by itself, raise both sides to the power of 4:

\[
x = \left( \frac{1}{2} \right)^4
\]

\[
x = \frac{1}{16}
\]

### Final Answer:
\[
x = \frac{1}{16}
\]

If you have any questions or want further details, feel free to ask!

Here are 5 related questions for practice:
1. Solve \( (3x^{\frac{1}{2}}) - (x^{\frac{1}{4}}) = 0 \).
2. Solve \( 4x^{\frac{1}{3}} = 16x^{\frac{1}{6}} \).
3. If \( y = x^{\frac{1}{2}} \), solve for \( x \) when \( y = 5 \).
4. Solve \( x^{\frac{1}{3}} - x^{\frac{1}{6}} = 0 \).
5. Simplify and solve \( x^{\frac{2}{3}} = 9 \).

**Tip:** When dealing with fractional exponents, remember that \( x^{m/n} \) is the same as \( \sqrt[n]{x^m} \), which helps in solving powers and roots.

Learn how to solve the equation involving fractional exponents: (2x^(1/2)) - (x^(1/4)) = 0. This step-by-step guide covers key algebraic concepts, including manipulating fractional powers and isolating terms. Suitable for students in Grades 9-11.

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