Exercise 2.1 — Class 9 Mathematics Solutions

Solution

1. Locate the decimal point and reference position.

   The reference position is after the first non-zero digit $2$.

2. Move the decimal point $6$ places to the left.

   Since the decimal moves $6$ places to the left, the power of $10$ is positive $+6$:

   $$\begin{aligned} 2000000 &= 2.000000 \times 10^6 \\ &= 2 \times 10^6 \end{aligned}$$

Solution

1. Identify the reference position.

   The first non-zero digit is $4$. The decimal point moves from the end of the number to right after $4$.

2. Count the decimal shifts.

   Moving $4$ places to the left gives a positive exponent of $+4$:

   $$\begin{aligned} 48900 &= 4.8900 \times 10^4 \\ &= 4.89 \times 10^4 \end{aligned}$$

Solution

1. Locate the first non-zero digit.

   The first non-zero digit is $4$. The decimal point must be placed after $4$.

2. Move the decimal point $3$ places to the right.

   Since the decimal moves to the right, the exponent of $10$ is negative ($-3$):

   $$\begin{aligned} 0.0042 &= 4.2 \times 10^{-3} \end{aligned}$$

Solution

1. Locate the first non-zero digit.

   The first non-zero digit is $9$.

2. Move the decimal point $7$ places to the right.

   Since the decimal point is shifted $7$ places to the right, the power of $10$ is $-7$:

   $$\begin{aligned} 0.0000009 &= 9 \times 10^{-7} \end{aligned}$$

Solution

1. Express $73$ in standard scientific form.

   $$73 = 7.3 \times 10^1$$

2. Combine powers of $10$ using the product rule $10^m \times 10^n = 10^{m+n}$.

   $$\begin{aligned} 73 \times 10^3 &= (7.3 \times 10^1) \times 10^3 \\ &= 7.3 \times 10^{1+3} \\ &= 7.3 \times 10^4 \end{aligned}$$

Solution

1. Express $0.65$ in standard scientific form.

   $$0.65 = 6.5 \times 10^{-1}$$

2. Combine powers of $10$.

   $$\begin{aligned} 0.65 \times 10^2 &= (6.5 \times 10^{-1}) \times 10^2 \\ &= 6.5 \times 10^{-1+2} \\ &= 6.5 \times 10^1 \end{aligned}$$

Solution

1. Identify the power of $10$.

   The power $10^2$ has a positive exponent $+2$, which equals $100$.

2. Shift the decimal point $2$ places to the right.

   $$\begin{aligned} 8.04 \times 10^2 &= 8.04 \times 100 \\ &= 804 \end{aligned}$$

Solution

1. Identify the power of $10$.

   The power is $10^5 = 100,000$.

2. Multiply by $3$.

   $$\begin{aligned} 3 \times 10^5 &= 3 \times 100,000 \\ &= 300,000 \end{aligned}$$

Solution

1. Identify the power of $10$.

   The exponent is $-2$, which represents $\frac{1}{10^2} = \frac{1}{100} = 0.01$.

2. Shift the decimal point $2$ places to the left.

   $$\begin{aligned} 1.5 \times 10^{-2} &= \frac{1.5}{100} \\ &= 0.015 \end{aligned}$$

Solution

1. Identify the power of $10$.

   The exponent is $+7$, which represents $10,000,000$.

2. Shift the decimal point $7$ places to the right.

   $$\begin{aligned} 1.77 \times 10^7 &= 1.77 \times 10,000,000 \\ &= 17,700,000 \end{aligned}$$

Solution

1. Identify the power of $10$.

   The exponent is $-6$, so move the decimal point $6$ places to the left.

2. Shift the decimal point.

   $$\begin{aligned} 5.5 \times 10^{-6} &= \frac{5.5}{1,000,000} \\ &= 0.0000055 \end{aligned}$$

Solution

1. Identify the power of $10$.

   The exponent is $-5$, so move the decimal point $5$ places to the left.

2. Shift the decimal point.

   $$\begin{aligned} 4 \times 10^{-5} &= \frac{4}{100,000} \\ &= 0.00004 \end{aligned}$$