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 $+6$:

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

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 $+4$:

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

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 ($-3$):

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

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 $-7$:

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

Solution

1. Express $73$ in standard scientific form.

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

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

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

Solution

1. Express $0.65$ in standard scientific form.

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

2. Combine powers of $10$.

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

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.

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

Solution

1. Identify the power of $10$.

   The power is ${10}^5=\mathrm{100,000}$.

2. Multiply by $3$.

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

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.

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

Solution

1. Identify the power of $10$.

   The exponent is $+7$, which represents $\mathrm{10,000,000}$.

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

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

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.

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

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.

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

Solution

1. Write the given scientific notation.

   Speed of light $=3\times {10}^8\text{ m/s}$

2. Expand into ordinary standard form.

   The exponent is $+8$, which equals $\mathrm{100,000,000}$:

   $$\begin{array}{cc}3\times {10}^8& =3\times \mathrm{100,000,000}\\ & =\mathrm{300,000,000}\text{ m/s}\end{array}$$

Solution

1. Identify the given value.

   Circumference of Earth $=\mathrm{40,075,000}\text{ metres}$

2. Move the decimal point to standard scientific position.

   The first non-zero digit is $4$. Moving the decimal point from the end to after $4$ requires shifting $7$ places to the left:

   $$\begin{array}{cc}40075000& =4.0075\times {10}^7\text{ metres}\end{array}$$

Solution

1. Write the given scientific notation.

   Diameter of Mars $=6.779\times {10}^3\text{ km}$

2. Convert into standard ordinary notation.

   The power is ${10}^3=1000$, so shift the decimal point $3$ places to the right:

   $$\begin{array}{cc}6.779\times {10}^3& =6.779\times 1000\\ & =6779\text{ km}\end{array}$$

Solution

1. Write the given scientific notation.

   Diameter of Earth $=1.2756\times {10}^4\text{ km}$

2. Convert into standard ordinary notation.

   The power is ${10}^4=\mathrm{10,000}$, so shift the decimal point $4$ places to the right:

   $$\begin{array}{cc}1.2756\times {10}^4& =1.2756\times \mathrm{10,000}\\ & =12756\text{ km}\end{array}$$