https://file.aiquickdraw.com/mathbot/file/1728568031466s1dphjad.jpg

The equation is:

\[
8x^{\frac{3}{2}} - x^{\frac{3}{2}} + 7 = 0
\]

Let's simplify and solve the equation:

1. Combine like terms:

\[
(8x^{\frac{3}{2}} - x^{\frac{3}{2}}) + 7 = 0
\]

This simplifies to:

\[
7x^{\frac{3}{2}} + 7 = 0
\]

2. Subtract 7 from both sides:

\[
7x^{\frac{3}{2}} = -7
\]

3. Divide both sides by 7:

\[
x^{\frac{3}{2}} = -1
\]

4. To solve for \(x\), raise both sides of the equation to the power of \(\frac{2}{3}\):

\[
x = (-1)^{\frac{2}{3}}
\]

The cube root of \(-1\) is \(-1\), and squaring that gives:

\[
x = 1
\]

Thus, the solution is \(x = 1\).

Would you like further details or have any questions?

Here are 5 follow-up questions to expand understanding:
1. How does raising both sides of an equation to a fractional power affect the solution?
2. Why can we combine the terms \(8x^{\frac{3}{2}}\) and \(-x^{\frac{3}{2}}\)?
3. What are the rules for solving equations involving fractional exponents?
4. How would the solution change if the exponent were different, say \(\frac{5}{2}\)?
5. How can we verify that \(x = 1\) satisfies the original equation?

**Tip:** Always check your final answer by substituting it back into the original equation to verify its correctness.

Solve the equation: 8x^(3/2) - x^(3/2) + 7 = 0

Learn how to solve the algebraic equation with fractional exponents: 8x^(3/2) - x^(3/2) + 7 = 0. This step-by-step solution covers simplifying expressions, combining like terms, and solving equations with fractional exponents. Suitable for students in Grades 9-12.

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