Question

Compare the following numbers \[4 \times {10^{14}};3 \times {10^{17}}\]

Answer 1

Hint:
We will compare the 2 numbers by expressing them in the same power of 10. We will use the formulas of exponents to express them as powers of 10. We can also divide the 2 numbers and check whether the result is less than 1, 1 or greater than 1.

Formulas used:
1) We know that any number say \[a\]to the power of \[b\] is the product of the number with itself \[b\] times:
\[{a^b} = \underbrace {a \times a \times a \times ... \times a}_{b{\rm{ times}}}\]
2) When we multiply 2 or more exponential numbers with the same base, their powers get added:
\[{a^p} \times {a^q} = {a^{p + q}}\]
3) When 2 exponential numbers with the same base are divided, their powers get subtracted:
\[\dfrac{{{a^p}}}{{{a^q}}} = {a^{p - q}}\]
4) The formula for a negative exponential power:
\[{a^{ - p}} = \dfrac{1}{{{a^p}}}\]

Complete step by step solution:
The first number we have is \[4 \times {10^{14}}\] and the second number is \[3 \times {10^{17}}\].
We will multiply and divide \[3 \times {10^{17}}\]by 1000:
\[\begin{array}{l} \Rightarrow 3 \times {10^{17}} = 3 \times {10^{17}} \times \dfrac{{1000}}{{1000}}\\ \Rightarrow 3 \times {10^{17}} = \dfrac{{3 \times {{10}^{17}} \times 1000}}{{10 \times 10 \times 10}}\end{array}\]
Substituting \[{10^3}\] for \[10 \times 10 \times 10\] in the denominator, we get
\[ \Rightarrow 3 \times {10^{17}} = \dfrac{{3 \times 1000 \times {{10}^{17}}}}{{{{10}^3}}}\]
Substituting 10 for \[a\], 17 for \[p\] and 3 for \[q\] in the formula \[\dfrac{{{a^p}}}{{{a^q}}} = {a^{p - q}}\]:
\[ \Rightarrow 3 \times {10^{17}} = 3000 \times {10^{17 - 3}}\]
\[ \Rightarrow 3 \times {10^{17}} = 3000 \times {10^{14}}\]
We know that
\[ \Rightarrow 3000 > 4\]
We will multiply both sides by \[{10^{14}}\] :
\[ \Rightarrow 3000 \times {10^{14}} > 4 \times {10^{14}}\]
We will substitute 10 for \[a\], 14 for \[p\] and 3 for \[q\] in the formula for multiplication of 2 exponential numbers with the same base on the left-hand side:
\[ \Rightarrow {\rm{ }}3 \times {10^{17}} > 4 \times {10^{14}}\]

$\therefore $ \[{\rm{ }}3 \times {10^{17}}\]is greater than \[4 \times {10^{14}}\].

Note:
We can also compare the numbers by dividing them. If the quotient is greater than 1, the numerator is greater than the denominator. If the quotient is equal to 1, the numerator and denominator are equal and if the quotient is less than 1, then the numerator is smaller than the denominator.
We will divide the 2 numbers:
\[\dfrac{{4 \times {{10}^{14}}}}{{3 \times {{10}^{17}}}}\]
Substituting 10 for \[a\], 14 for \[p\] and 17 for \[q\] in the formula \[\dfrac{{{a^p}}}{{{a^q}}} = {a^{p - q}}\], we get
\[\begin{array}{l} \Rightarrow \dfrac{{4 \times {{10}^{14}}}}{{3 \times {{10}^{17}}}} = \dfrac{{4 \times {{10}^{14 - 17}}}}{3}\\ \Rightarrow \dfrac{{4 \times {{10}^{14}}}}{{3 \times {{10}^{17}}}} = \dfrac{4}{3} \times {10^{ - 3}}\end{array}\]
Substituting 10 for \[a\]and 3 for \[p\] in the formula \[{a^{ - p}} = \dfrac{1}{{{a^p}}}\], we get
\[\begin{array}{l} \Rightarrow \dfrac{{4 \times {{10}^{14}}}}{{3 \times {{10}^{17}}}} = \dfrac{4}{3} \times \dfrac{1}{{{{10}^3}}}\\ \Rightarrow \dfrac{{4 \times {{10}^{14}}}}{{3 \times {{10}^{17}}}} = \dfrac{{1.333}}{{{{10}^3}}}\\ \Rightarrow \dfrac{{4 \times {{10}^{14}}}}{{3 \times {{10}^{17}}}} = 0.001333\end{array}\]
We can see that \[0.001333 < 1\].
 $\therefore $ \[4 \times {10^{14}}\] is less than \[3 \times {10^{17}}\].