5) $$ 3 x^{-\frac{3}{2}}-1=80 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$ 3x^{-\frac{3}{2}} - 1 = 80 $$

follow these steps:

1. **Isolate the Term with $$ x $$:**

Start by adding 1 to both sides of the equation to isolate the term containing $$ x $$:

$$
   3x^{-\frac{3}{2}} = 81
   $$

2. **Solve for $$ x^{-\frac{3}{2}} $$:**

Divide both sides by 3 to simplify:

$$
   x^{-\frac{3}{2}} = \frac{81}{3} = 27
   $$

3. **Rewrite the Equation:**

Recall that $$ x^{-\frac{3}{2}} $$ is the same as $$ \frac{1}{x^{\frac{3}{2}}} $$. So,

$$
   \frac{1}{x^{\frac{3}{2}}} = 27
   $$

Taking the reciprocal of both sides gives:

$$
   x^{\frac{3}{2}} = \frac{1}{27}
   $$

4. **Solve for $$ x $$:**

To solve for $$ x $$, raise both sides of the equation to the power of $$ \frac{2}{3} $$. This is because:

$$
   \left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = x^{\frac{3}{2} \times \frac{2}{3}} = x^1 = x
   $$

Applying this to both sides:

$$
   x = \left(\frac{1}{27}\right)^{\frac{2}{3}}
   $$

5. **Simplify the Right Side:**

Break down $$ 27 $$ as $$ 3^3 $$:

$$
   x = \left(\frac{1}{3^3}\right)^{\frac{2}{3}} = \left(3^{-3}\right)^{\frac{2}{3}} = 3^{-2} = \frac{1}{9}
   $$

6. **Final Solution:**

$$
   x = \frac{1}{9}
   $$

**Therefore, the solution to the equation is $$ x = \frac{1}{9} $$.**
$$ x = \frac{1}{9} $$

5) \( 3 x^{-\frac{3}{2}}-1=80 \)