10) $$ 2(4 x-3)^{\frac{5}{4}}+9=495 $$

Question

UpStudy AI · Accepted Answer

Let's solve the equation step by step:

$$
2(4x - 3)^{\frac{5}{4}} + 9 = 495
$$

### Step 1: Isolate the Exponential Term

Subtract 9 from both sides to isolate the term with the exponent:

$$
2(4x - 3)^{\frac{5}{4}} = 495 - 9
$$
$$
2(4x - 3)^{\frac{5}{4}} = 486
$$

### Step 2: Divide Both Sides by 2

$$
(4x - 3)^{\frac{5}{4}} = \frac{486}{2}
$$
$$
(4x - 3)^{\frac{5}{4}} = 243
$$

### Step 3: Eliminate the Exponent

To remove the exponent $$\frac{5}{4}$$, raise both sides of the equation to the reciprocal of $$\frac{5}{4}$$, which is $$\frac{4}{5}$$:

$$
\left[(4x - 3)^{\frac{5}{4}}\right]^{\frac{4}{5}} = 243^{\frac{4}{5}}
$$
$$
4x - 3 = 243^{\frac{4}{5}}
$$

### Step 4: Simplify the Right Side

Recognize that $$243$$ is a power of $$3$$:

$$
243 = 3^5
$$
$$
243^{\frac{4}{5}} = (3^5)^{\frac{4}{5}} = 3^{4} = 81
$$

So,

$$
4x - 3 = 81
$$

### Step 5: Solve for $$x$$

Add 3 to both sides:

$$
4x = 81 + 3
$$
$$
4x = 84
$$

Divide both sides by 4:

$$
x = \frac{84}{4}
$$
$$
x = 21
$$

### **Solution:**
$$
x = 21
$$
$$ x = 21 $$

10) \( 2(4 x-3)^{\frac{5}{4}}+9=495 \)