how do I solve equations with fractional powers?

To solve equations with fractional powers, the key idea is to isolate the variable and then eliminate the fractional exponent. Here's a step-by-step guide:

### General Approach:
1. **Isolate the term with the fractional exponent**: Move all other terms to the other side of the equation, leaving the term with the fractional power on one side.

   Example:
   \[
   x^{\frac{3}{2}} = 8
   \]

2. **Raise both sides to the reciprocal of the fractional exponent**: The goal is to eliminate the fractional exponent by raising both sides of the equation to the reciprocal of the fraction. The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

   For example, if the exponent is \(\frac{3}{2}\), raise both sides to the power of \(\frac{2}{3}\):
   \[
   \left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 8^{\frac{2}{3}}
   \]
   This simplifies to:
   \[
   x = 8^{\frac{2}{3}}
   \]

3. **Simplify the right-hand side**: Evaluate the expression on the right-hand side by applying the power and/or root.
   
   In this case, first, take the cube root of 8, which is 2:
   \[
   8^{\frac{1}{3}} = 2
   \]
   Then square 2:
   \[
   2^2 = 4
   \]
   So, \(x = 4\).

4. **Check for extraneous solutions**: If you're solving an equation where the exponent results in taking an even root (like square roots), you may need to check if both positive and negative values work. For example, if you're solving \(x^{\frac{1}{2}} = 3\), then both \(x = 9\) and \(x = -9\) should be checked.

### Example Problem:
Solve the equation:
\[
x^{\frac{3}{4}} = 16
\]

#### Step 1: Raise both sides to the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\):
\[
\left(x^{\frac{3}{4}}\right)^{\frac{4}{3}} = 16^{\frac{4}{3}}
\]

#### Step 2: Simplify:
\[
x = 16^{\frac{4}{3}}
\]
Take the cube root of 16:
\[
16^{\frac{1}{3}} \approx 2.52
\]
Then raise it to the fourth power:
\[
x \approx 2.52^4 \approx 40.31
\]

So, \(x \approx 40.31\).

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Would you like to try solving a specific equation with fractional powers or need more detailed steps?

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### Relative Questions:
1. How do you solve equations with negative fractional exponents?
2. What if the fractional exponent is negative, how does that change the steps?
3. Can fractional powers be solved graphically, and how?
4. What happens if the equation has more than one term with fractional powers?
5. How do fractional powers relate to logarithms?

**Tip**: Remember, fractional exponents represent roots, so \(\frac{1}{2}\) means square root, \(\frac{1}{3}\) means cube root, and so on.

Learn how to solve equations with fractional powers, such as x^(3/2) = 8, using reciprocal exponents and simplifying roots. This detailed guide explains key algebraic concepts, suitable for students in Grades 9-11.

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