Question

Solve the equation and check your solution

√5x+1=6

-2√24x+13= - 11

√2x - 2/3=0

2, 5 exponent √x + 7 = 15

³√x - 16 = 2

³√x-10= - 7

8³√10x - 15 = 17

⁴√4x - 13 = - 15

Answer 1

Let's solve each equation step by step and check the solutions.

### 1. Solve \( \sqrt{5x + 1} = 6 \)

**Step 1:** Square both sides to eliminate the square root.
\[
5x + 1 = 36
\]

**Step 2:** Subtract 1 from both sides.
\[
5x = 35
\]

**Step 3:** Divide by 5.
\[
x = 7
\]

**Check:**
\[
\sqrt{5(7) + 1} = \sqrt{35 + 1} = \sqrt{36} = 6 \quad \text{(Correct)}
\]

### 2. Solve \( -2\sqrt{24x + 13} = -11 \)

**Step 1:** Divide by -2.
\[
\sqrt{24x + 13} = \frac{11}{2}
\]

**Step 2:** Square both sides.
\[
24x + 13 = \left(\frac{11}{2}\right)^2 = \frac{121}{4}
\]

**Step 3:** Multiply everything by 4 to eliminate the fraction.
\[
96x + 52 = 121
\]

**Step 4:** Subtract 52 from both sides.
\[
96x = 69
\]

**Step 5:** Divide by 96.
\[
x = \frac{69}{96} = \frac{23}{32}
\]

**Check:**
\[
-2\sqrt{24\left(\frac{23}{32}\right) + 13} = -2\sqrt{\frac{69}{4} + 13} = -2\sqrt{\frac{69 + 52}{4}} = -2\sqrt{\frac{121}{4}} = -2 \cdot \frac{11}{2} = -11 \quad \text{(Correct)}
\]

### 3. Solve \( \sqrt{2x} - \frac{2}{3} = 0 \)

**Step 1:** Add \( \frac{2}{3} \) to both sides.
\[
\sqrt{2x} = \frac{2}{3}
\]

**Step 2:** Square both sides.
\[
2x = \left(\frac{2}{3}\right)^2 = \frac{4}{9}
\]

**Step 3:** Divide by 2.
\[
x = \frac{4}{18} = \frac{2}{9}
\]

**Check:**
\[
\sqrt{2\left(\frac{2}{9}\right)} - \frac{2}{3} = \sqrt{\frac{4}{9}} - \frac{2}{3} = \frac{2}{3} - \frac{2}{3} = 0 \quad \text{(Correct)}
\]

### 4. Solve \( 2 \cdot 5^{\sqrt{x}} + 7 = 15 \)

**Step 1:** Subtract 7 from both sides.
\[
2 \cdot 5^{\sqrt{x}} = 8
\]

**Step 2:** Divide by 2.
\[
5^{\sqrt{x}} = 4
\]

**Step 3:** Rewrite 4 as \(5^{\log_5(4)}\) and take logarithm.
\[
\sqrt{x} = \log_5(4)
\]

**Step 4:** Square both sides.
\[
x = (\log_5(4))^2
\]

**Check:** This step doesn't yield a simple check, so we'll leave it as \( x = (\log_5(4))^2 \).

### 5. Solve \( \sqrt[3]{x} - 16 = 2 \)

**Step 1:** Add 16 to both sides.
\[
\sqrt[3]{x} = 18
\]

**Step 2:** Cube both sides.
\[
x = 18^3 = 5832
\]

**Check:**
\[
\sqrt[3]{5832} - 16 = 18 - 16 = 2 \quad \text{(Correct)}
\]

### 6. Solve \( \sqrt[3]{x} - 10 = -7 \)

**Step 1:** Add 10 to both sides.
\[
\sqrt[3]{x} = 3
\]

**Step 2:** Cube both sides.
\[
x = 3^3 = 27
\]

**Check:**
\[
\sqrt[3]{27} - 10 = 3 - 10 = -7 \quad \text{(Correct)}
\]

### 7. Solve \( 8\sqrt[3]{10x} - 15 = 17 \)

**Step 1:** Add 15 to both sides.
\[
8\sqrt[3]{10x} = 32
\]

**Step 2:** Divide by 8.
\[
\sqrt[3]{10x} = 4
\]

**Step 3:** Cube both sides.
\[
10x = 64
\]

**Step 4:** Divide by 10.
\[
x = 6.4
\]

**Check:**
\[
8\sqrt[3]{10(6.4)} - 15 = 8\sqrt[3]{64} - 15 = 8 \cdot 4 - 15 = 32 - 15 = 17 \quad \text{(Correct)}
\]

### 8. Solve \( \sqrt[4]{4x} - 13 = -15 \)

**Step 1:** Add 13 to both sides.
\[
\sqrt[4]{4x} = -2
\]

Since the fourth root cannot be negative, there are no solutions for this equation.

### Summary of Solutions:
1. \( x = 7 \)
2. \( x = \frac{23}{32} \)
3. \( x = \frac{2}{9} \)
4. \( x = (\log_5(4))^2 \)
5. \( x = 5832 \)
6. \( x = 27 \)
7. \( x = 6.4 \)
8. No solution.