Find the GM between the numbers
(i) 5 and 125 (ii) 1 and `9/16` (iii) 0.15 and 0.0015
(iv) -8 and -2 (v) -6.3 and -2.8 (v) `a^(3)b` and `ab^(3)`

Answer

To find the geometric mean (GM) between two numbers, we use the formula:

\[ \text{GM} = \sqrt{a \times b} \]

where \(a\) and \(b\) are the two numbers. Let's solve each part step by step.

### (i) Find the GM between 5 and 125

1. **Multiply the two numbers**:
\[ 5 \times 125 = 625 \]

2. **Take the square root**:
\[ \text{GM} = \sqrt{625} = 25 \]

### (ii) Find the GM between 1 and \( \frac{9}{16} \)

1. **Multiply the two numbers**:
\[ 1 \times \frac{9}{16} = \frac{9}{16} \]

2. **Take the square root**:
\[ \text{GM} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### (iii) Find the GM between 0.15 and 0.0015

1. **Multiply the two numbers**:
\[ 0.15 \times 0.0015 = 0.000225 \]

2. **Take the square root**:
\[ \text{GM} = \sqrt{0.000225} = 0.015 \]

### (iv) Find the GM between -8 and -2

1. **Multiply the two numbers**:
\[ -8 \times -2 = 16 \]

2. **Take the square root**:
\[ \text{GM} = \sqrt{16} = 4 \]

### (v) Find the GM between -6.3 and -2.8

1. **Multiply the two numbers**:
\[ -6.3 \times -2.8 = 17.64 \]

2. **Take the square root**:
\[ \text{GM} = \sqrt{17.64} \approx 4.2 \]

### (vi) Find the GM between \( a^3b \) and \( ab^3 \)

1. **Multiply the two expressions**:
\[ a^3b \times ab^3 = a^{3+1}b^{1+3} = a^4b^4 \]

2. **Take the square root**:
\[ \text{GM} = \sqrt{a^4b^4} = a^2b^2 \]

### Summary of Results:
- (i) GM of 5 and 125 is **25**
- (ii) GM of 1 and \( \frac{9}{16} \) is **\( \frac{3}{4} \)**
- (iii) GM of 0.15 and 0.0015 is **0.015**
- (iv) GM of -8 and -2 is **4**
- (v) GM of -6.3 and -2.8 is **approximately 4.2**
- (vi) GM of \( a^3b \) and \( ab^3 \) is **\( a^2b^2 \)**