The number of solutions of `(81)^(sin ^2 x) + (81)^( cos^2x )=30` for `x in [0,2pi]` is equal to…………………

Answer

To solve the equation \( (81)^{\sin^2 x} + (81)^{\cos^2 x} = 30 \) for \( x \) in the interval \([0, 2\pi]\), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \sin^2 x + \cos^2 x = 1 \). Therefore, we can express \( \cos^2 x \) as \( 1 - \sin^2 x \).

Substituting this into the equation gives us:
\[ (81)^{\sin^2 x} + (81)^{1 - \sin^2 x} = 30 \]

### Step 2: Simplify the equation
Let \( y = (81)^{\sin^2 x} \). Then, we can rewrite the equation as:
\[ y + \frac{81}{y} = 30 \]

### Step 3: Multiply through by \( y \)
To eliminate the fraction, multiply both sides by \( y \):
\[ y^2 + 81 = 30y \]

### Step 4: Rearrange into standard quadratic form
Rearranging gives us:
\[ y^2 - 30y + 81 = 0 \]

### Step 5: Solve the quadratic equation
Now we can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -30, c = 81 \):
\[ y = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 1 \cdot 81}}{2 \cdot 1} \]
\[ y = \frac{30 \pm \sqrt{900 - 324}}{2} \]
\[ y = \frac{30 \pm \sqrt{576}}{2} \]
\[ y = \frac{30 \pm 24}{2} \]

Calculating the two possible values for \( y \):
1. \( y = \frac{54}{2} = 27 \)
2. \( y = \frac{6}{2} = 3 \)

### Step 6: Substitute back for \( \sin^2 x \)
Recall that \( y = (81)^{\sin^2 x} \). Thus:
1. For \( y = 27 \):
\[ (81)^{\sin^2 x} = 27 \implies 3^{4\sin^2 x} = 3^3 \implies 4\sin^2 x = 3 \implies \sin^2 x = \frac{3}{4} \]
Therefore, \( \sin x = \pm \frac{\sqrt{3}}{2} \).

2. For \( y = 3 \):
\[ (81)^{\sin^2 x} = 3 \implies 3^{4\sin^2 x} = 3^1 \implies 4\sin^2 x = 1 \implies \sin^2 x = \frac{1}{4} \]
Therefore, \( \sin x = \pm \frac{1}{2} \).

### Step 7: Find solutions for \( \sin x = \pm \frac{\sqrt{3}}{2} \)
- \( \sin x = \frac{\sqrt{3}}{2} \) gives \( x = \frac{\pi}{3}, \frac{2\pi}{3} \).
- \( \sin x = -\frac{\sqrt{3}}{2} \) gives \( x = \frac{4\pi}{3}, \frac{5\pi}{3} \).

### Step 8: Find solutions for \( \sin x = \pm \frac{1}{2} \)
- \( \sin x = \frac{1}{2} \) gives \( x = \frac{\pi}{6}, \frac{5\pi}{6} \).
- \( \sin x = -\frac{1}{2} \) gives \( x = \frac{7\pi}{6}, \frac{11\pi}{6} \).

### Step 9: Count the total number of solutions
From both cases, we have:
- From \( \sin x = \pm \frac{\sqrt{3}}{2} \): 4 solutions.
- From \( \sin x = \pm \frac{1}{2} \): 4 solutions.

Thus, the total number of solutions is \( 4 + 4 = 8 \).

### Final Answer
The number of solutions of the equation \( (81)^{\sin^2 x} + (81)^{\cos^2 x} = 30 \) for \( x \) in the interval \([0, 2\pi]\) is **8**.

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