DAV Class 7 Maths Chapter 4 Worksheet 7
Exponents and Powers Worksheet 7
1. Write each of the following numbers in the form \( k \times 10^n \) where \( 1\leq k < 10 \) and \( n \) is an integer.
(i) \( 1,384,000 \)
Solution
\begin{align*} & = 1,384,000 \\ & = \color{red} 1.384 \times 10^6 \end{align*}
(ii) \( 12.32005 \)
Solution
\begin{align*} & = 12.32005 \\ & = \color{red} 1.232005 \times 10^1 \end{align*}
(iii) \( 2157.957 \)
Solution
\begin{align*} & = 2157.957 \\ & = \color{red} 2.157957 \times 10^3 \end{align*}
(iv) \( 0.00002 \)
Solution
\begin{align*} & = 0.00002 \\ & = \color{red} 2.0 \times 10^{-5} \end{align*}
(v) \( 0.00729 \)
Solution
\begin{align*} & = 0.00729 \\ & = \color{red} 7.29 \times 10^{-3} \end{align*}
(vi) \( 0.000000000926 \)
Solution
\begin{align*} & = 0.000000000926 \\ & = \color{red} 9.26 \times 10^{-10} \end{align*}
(vii) \( 520,000,000 \)
Solution
\begin{align*} & = 520,000,000 \\ & = \color{red} 5.2 \times 10^8 \end{align*}
(viii) \( 0.0000085 \)
Solution
\begin{align*} & = 0.0000085 \\ & = \color{red} 8.5 \times 10^{-6} \end{align*}
2. Write the following numbers in the usual form.
(i) \( 52.5 \times 10^4 \)
Solution
\begin{align*} &= 52.5 \times 10^4 \\ &= 52.5 \times 10000 \\ &= \color{red} 525000 \end{align*}
(ii) \( 158.9 \times 10^6 \)
Solution
\begin{align*} &= 158.9 \times 10^6 \\ &= 158.9 \times 1000000 \\ &= \color{red} 158900000 \end{align*}
(iii) \( 9.545 \times 10^{12} \)
Solution
\begin{align*} &= 9.545 \times 10^{12} \\ &= 9.545 \times 1000000000000 \\ &= \color{red} 9545000000000 \end{align*}
(iv) \( 1.72 \times 10^{-5} \)
Solution
\begin{align*} & = 1.72 \times 10^{-5} \\ \\ &= 1.72 \times \frac{1}{10^{5}} \\ \\ &= 1.72 \times \frac{1}{100000} \\ \\ &= \color{red} 0.0000172 \end{align*}
(v) \( 8.5 \times 10^{-7} \)
Solution
\begin{align*} & = 8.5 \times 10^{-7} \\ \\ &= 8.5 \times \frac{1}{10^{7}} \\ \\ &= 8.5 \times \frac{1}{10000000} \\ \\ &= \color{red} 0.00000085 \end{align*}
(vi) \( 2.9 \times 10^{-9} \)
Solution
\begin{align*} & = 2.9 \times 10^{-9} \\ \\ &= 2.9 \times \frac{1}{10^{9}} \\ \\ &= 2.9 \times \frac{1}{1000000000} \\ \\ &= \color{red} 0.0000000029 \end{align*}
(vii) \( 6293.2 \times 10^5 \)
Solution
\begin{align*} &= 6293.2 \times 10^5 \\ &= 6293.2 \times 100000 \\ &= \color{red} 629320000 \end{align*}
(viii) \( 1.925 \times 10^{-6} \)
Solution
\begin{align*} & = 1.925 \times 10^{-6} \\ \\ &= 1.925 \times \frac{1}{10^{6}} \\ \\ &= 1.925 \times \frac{1}{1000000} \\ \\ &= \color{red} 0.000001925 \end{align*}
3. Express each of the following in the form \( k \times 10^n ; (1 ≤ k < 10) \)
(i) \( (1.25 \times 10^7) \div (5 \times 10^3) \)
Solution
\begin{align*} & = (1.25 \times 10^7) \div (5 \times 10^3) \\ \\ &= \frac{1.25 \times 10^7}{5 \times 10^3} \\ \\ &= \frac{\cancel{1.25}^{0.25} }{\cancel5_1} \times \frac{10^7}{10^3} \\ \\ &= 0.25 \times 10^{7-3} \\ &= 0.25 \times 10^{4} \\ &= \color{red} 2.5 \times 10^3 \end{align*}
(ii) \((2.5 \times 10^{10}) \times (31.25 \times 10^{-5})\)
Solution
\begin{align*} &= 2.5 \times 10^{10} \times 31.25 \times 10^{-5}\\ &= 2.5 \times 31.25 \times 10^{10} \times 10^{-5}\\ &= 78.125 \times 10^{10 + (-5)}\\ &= 78.125 \times 10^{10 - 5}\\ &= 78.125 \times 10^{5}\\ &= \color{red} 7.8125 \times 10^{6}\\ \end{align*}
(iii) \((1.6 \times 10^{9}) \times (5.0 \times 10^{-3})\)
Solution
\begin{align*} &= 1.6 \times 10^{9} \times 5 \times 10^{-3}\\ &= 1.6 \times 5 \times 10^{9} \times 10^{-3}\\ &= 8 \times 10^{9 + (-3)}\\ &= 8 \times 10^{9 - 3}\\ &= \color{red} 8 \times 10^{6}\\ \end{align*}
(iv) \( [(3.4 \times 10^{4}) \times (5 \times 10^{-3})] \div [2.0 \times 10^5]\)
Solution
\begin{align*} &= \frac{(3.4 \times 10^{4}) \times (5 \times 10^{-3})}{2.0 \times 10^5} \\ \\ &= \frac{\cancel{3.4}^{1.7} \times 5 \times 10^{4} \times 10^{-3}}{\cancel 2_1 \times 10^5} \\ \\ &= 1.7 \times 5 \times 10^{4+(-3)-5} \\ &= 8.5 \times 10^{4-3-5} \\ &= 8.5 \times 10^{4-8} \\ &= \color{red} 8.5 \times 10^{-4} \end{align*}
4. Express the numbers appearing in the following statements in the form \( k \times 10^n \) where \( (1 ≤ k < 10) \) and \( n \) is an integer.
(i) Sun's diameter is \(1,384,000 \) km.
Solution
\begin{align*} &= 1,384,000 \\ &= 1.384000 \times 10^6 \\ &= \color{red} 1.384 \times 10^6 \ km \\ \end{align*}
(ii) The distance of the sun from the earth is approximately \(150,000,000 \ km \).
Solution
\begin{align*} &= 150,000,000 \\ &= 1.50000000 \times 10^8 \\ &= \color{red} 1.5 \times 10^8 \ km \\ \end{align*}
(iii) The speed of the light is about \(27600 \, \text{km/sec.}\)
Solution
\begin{align*} &= 27600 \\ &= 2.7600 \times 10^4 \\ &= \color{red} 2.76 \times 10^4 \ km/sec \\ \end{align*}
(iv) 1 Angstrom unit = \(\frac{1}{10,000,000,000} \, m\).
Solution
\begin{align*} &= \frac{1}{10,000,000,000} \\ \\ &= \frac{1}{10^{10}} \\ \\ &= \color{red} 1.0 \times 10^{-10} \ m\\ \end{align*}