Solve for x x^(3/2)-8=0

Solution:

Step 1:

Isolate the variable term by adding $8$ to each side of the original equation: $x^{\frac{3}{2}} = 8$.

Step 2:

To remove the fractional exponent, raise both sides to the reciprocal power, which is $\frac{2}{3}$: $\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 8^{\frac{2}{3}}$.

Step 3:

Proceed to simplify the exponents.

Step 3.1:

Start by simplifying the left-hand side.

Step 3.1.1:

Apply the exponent rule to $\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}}$.

Step 3.1.1.1:

Multiply the exponents according to the power of a power rule: $x^{\frac{3}{2} \cdot \frac{2}{3}} = 8^{\frac{2}{3}}$.

Step 3.1.1.1.1:

Eliminate the common factor of $3$ in the exponents: $x^{\frac{\cancel{3}}{2} \cdot \frac{2}{\cancel{3}}} = 8^{\frac{2}{3}}$.

Step 3.1.1.1.2:

Rewrite the simplified expression: $x^{\frac{1}{2} \cdot 2} = 8^{\frac{2}{3}}$.

Step 3.1.1.1.3:

Further simplify by canceling the common factor of $2$: $x^{\frac{1}{\cancel{2}} \cdot \cancel{2}} = 8^{\frac{2}{3}}$.

Step 3.1.1.1.3.1:

Conclude the simplification of the left-hand side: $x = 8^{\frac{2}{3}}$.

Step 3.2:

Now, simplify the right-hand side.

Step 3.2.1:

Focus on simplifying $8^{\frac{2}{3}}$.

Step 3.2.1.1:

Express $8$ as a power of $2$: $x = \left(2^{3}\right)^{\frac{2}{3}}$.

Step 3.2.1.1.1:

Apply the power rule by multiplying the exponents: $x = 2^{3 \cdot \frac{2}{3}}$.

Step 3.2.1.2:

Eliminate the common factor of $3$: $x = 2^{\cancel{3} \cdot \frac{2}{\cancel{3}}}$.

Step 3.2.1.2.1:

Write down the simplified expression: $x = 2^{2}$.

Step 3.2.1.3:

Calculate $2$ to the power of $2$: $x = 4$.

The final solution is $x = 4$.

Knowledge Notes:

The problem-solving process involves solving an equation with a fractional exponent. Here are the relevant knowledge points:

1. Isolating the Variable Term: The first step in solving an equation is to isolate the variable term on one side of the equation.

2. Fractional Exponents: A fractional exponent, such as $x^{\frac{a}{b}}$, can be interpreted as the $b$-th root of $x$ raised to the $a$-th power.

3. Inverse Operations: To eliminate a fractional exponent, we can use the inverse operation. In this case, raising both sides to the power of $\frac{2}{3}$ is the inverse operation of the $\frac{3}{2}$ exponent.

4. Power of a Power Rule: When an exponent is raised to another exponent, you multiply the exponents. This is expressed as $(a^{m})^{n} = a^{m \cdot n}$.

5. Simplifying Exponents: When simplifying expressions with exponents, common factors in the numerators and denominators can be canceled out.

6. Evaluating Powers: The final step often involves evaluating powers of numbers to find the solution to the equation.

Understanding these concepts is crucial for solving equations involving exponents and for algebraic manipulation in general.