Question

Simplify $\dfrac{{1.5 \times {{10}^6}}}{{2.5 \times {{10}^{ - 4}}}}$ and express in the standard form.

Answer 1

Hint:
We can simplify the powers of 10 by multiplying with the same terms on both numerator and denominators. Then we can simplify further using properties of exponents. Then we can make the decimals to whole numbers and we can divide them. Then we can make it in the standard form by using the properties of exponents.

Complete step by step solution:
We have the expression $\dfrac{{1.5 \times {{10}^6}}}{{2.5 \times {{10}^{ - 4}}}}$.
Let $I = \dfrac{{1.5 \times {{10}^6}}}{{2.5 \times {{10}^{ - 4}}}}$.
We can multiply both numerator and denominator with ${10^4}$. So, we will get,
 $ \Rightarrow I = \dfrac{{1.5 \times {{10}^6} \times {{10}^4}}}{{2.5 \times {{10}^{ - 4}} \times {{10}^4}}}$
We know that, ${a^x} \times {a^y} = {a^{x + y}}$. On applying this on both numerator and denominator, we get,
 $ \Rightarrow I = \dfrac{{1.5 \times {{10}^{6 + 4}}}}{{2.5 \times {{10}^{ - 4 + 4}}}}$
We can calculate the sum in the powers,
 $ \Rightarrow I = \dfrac{{1.5 \times {{10}^{10}}}}{{2.5 \times {{10}^0}}}$
We know that ${a^0} = 1$ . So, the denominator will become,
 $ \Rightarrow I = \dfrac{{1.5 \times {{10}^{10}}}}{{2.5 \times 1}}$
Now we know that ${a^n} = a \times {a^{n - 1}}$ . We can use this in the numerator
 $ \Rightarrow I = \dfrac{{1.5 \times 10 \times {{10}^9}}}{{2.5}}$
Now we can divide. We know that $\dfrac{{10}}{{2.5}} = 4$. So, we get,
 $ \Rightarrow I = 1.5 \times 4 \times {10^9}$
On multiplication, we get,
 $ \Rightarrow I = 6 \times {10^9}$

Therefore, the given expression in its standard form is given by $6 \times {10^9}$.

Note:
Alternate method for solving this problem is given by,
We have the expression $\dfrac{{1.5 \times {{10}^6}}}{{2.5 \times {{10}^{ - 4}}}}$ .
Let $I = \dfrac{{1.5 \times {{10}^6}}}{{2.5 \times {{10}^{ - 4}}}}$ .
We can multiply both numerator and denominator with 10. So, we will get,
 \[ \Rightarrow I = \dfrac{{1.5 \times 10 \times {{10}^6}}}{{2.5 \times 10 \times {{10}^{ - 4}}}}\]
Now we can multiply the 1st two terms in both numerator and denominator. So, we will get,
 \[ \Rightarrow I = \dfrac{{15 \times {{10}^6}}}{{25 \times {{10}^{ - 4}}}}\]
Now we know that ${a^n} = a \times {a^{n - 1}}$. We can use this in the numerator,
 \[ \Rightarrow I = \dfrac{{15 \times 10 \times {{10}^5}}}{{25 \times {{10}^{ - 4}}}}\]
 \[ \Rightarrow I = \dfrac{{150 \times {{10}^5}}}{{25 \times {{10}^{ - 4}}}}\]
Now we can divide the whole numbers. So, we will get,
 \[ \Rightarrow I = \dfrac{{6 \times {{10}^5}}}{{{{10}^{ - 4}}}}\]
We know that, $\dfrac{{{a^x}}}{{{a^y}}} = {a^{x - y}}$. So, on applying this, we get,
 \[ \Rightarrow I = 6 \times {10^{5 - ( - 4)}}\]
On simplification of the powers, we get.
 \[ \Rightarrow I = 6 \times {10^{5 + 4}}\]
Hence, we have,
 \[ \Rightarrow I = 6 \times {10^9}\]
Therefore, the given expression in its standard form is given by $6 \times {10^9}$.