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what will be the answer if x=14,y=2,z=7 And a=32

Let's solve the given equations step by step using the provided values \(x = 14\), \(y = 2\), \(z = 7\), and \(a = 32\).

### Equation 1:
\[
\sqrt{z - \frac{2}{x}} = 2\sqrt{\frac{y}{z + 1}}
\]

Substituting the given values:
\[
\sqrt{7 - \frac{2}{14}} = 2\sqrt{\frac{2}{7 + 1}}
\]

Simplify:
\[
\sqrt{7 - \frac{1}{7}} = 2\sqrt{\frac{2}{8}} 
\]
\[
\sqrt{\frac{49 - 1}{7}} = 2\sqrt{\frac{1}{4}} 
\]
\[
\sqrt{\frac{48}{7}} = 2 \times \frac{1}{2}
\]
\[
\sqrt{\frac{48}{7}} = 1
\]

\[
\sqrt{\frac{48}{7}} \approx 2.61 \neq 1 
\]

So, the given values do not satisfy Equation 1.

### Equation 2:
\[
y^x = z \times y + x \times y
\]

Substituting the values:
\[
2^{14} = 7 \times 2 + 14 \times 2
\]

Simplify:
\[
2^{14} = 14 + 28
\]
\[
16384 = 42
\]

This equation is also not satisfied.

### Equation 3:
\[
\log_a (x) + \log_y (z) = \log_z (a)
\]

Substituting the values:
\[
\log_{32} (14) + \log_2 (7) = \log_7 (32)
\]

These logarithms can be calculated as:
\[
\log_{32} (14) \approx 0.7834, \quad \log_2 (7) \approx 2.8074, \quad \log_7 (32) \approx 1.7386
\]

\[
0.7834 + 2.8074 \neq 1.7386
\]

So, this equation is also not satisfied.

### Integral Expression:
\[
\int_{y^x}^{z+m\cdot \sin(x)} \, dt = \text{What will be the answer?}
\]

Substituting the values:
\[
\int_{2^{14}}^{7 + m\cdot \sin(14)} \, dt
\]

The integral of a constant \( dt \) is just the difference between the upper and lower limits:
\[
= \text{upper limit} - \text{lower limit} = \left(7 + m\cdot \sin(14)\right) - 2^{14}
\]

\[
= 7 + m \cdot \sin(14) - 16384
\]

Without the specific value of \( m \), this expression remains in terms of \( m \).

### Summary:
- Equation 1: Not satisfied.
- Equation 2: Not satisfied.
- Equation 3: Not satisfied.
- Integral: \( 7 + m \cdot \sin(14) - 16384 \).

Would you like more details on any of these steps or have any questions?

**Related Questions:**
1. How do you solve logarithmic equations in different bases?
2. What are the steps to simplify square root equations?
3. How do you evaluate definite integrals with trigonometric functions?
4. Can you explain the properties of logarithms that were used here?
5. What are common pitfalls in solving algebraic equations?

**Tip:** Always double-check the values you substitute into equations to ensure accurate calculations.

Explore how to solve equations involving logarithms, integrals, and constants step-by-step with detailed explanations. This problem includes specific values for variables x, y, z, and a, providing insights into mathematical problem-solving strategies. Suitable for advanced high school students and learners interested in mathematical equations.

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