Question

Look at the pattern given below
12 × 5 = 12 × $\dfrac{10}{2}$ = $\dfrac{120}{2}$ = 60.
12 × 25 = 12 × $\dfrac{100}{4}$ = $\dfrac{1200}{4}=300$
Now find the value of $\text{a) 12 }\!\!\times\!\!\text{ 125}$ and $\text{b) 12}\times \text{625}$

Answer 1

Hint: Now we are given with two equation which are 12 × 5 = 12 × $\dfrac{10}{2}$ = $\dfrac{120}{2}$ = 60.
12 × 25 = 12 × $\dfrac{100}{4}$ = $\dfrac{1200}{4}=300$ . Now if we note we are multiplying powers of 5 to 12. As in first case ${{5}^{1}}$ and in second case ${{5}^{2}}$ and in result we are getting something in the form of $\dfrac{12\times ({{10}^{n}})}{4}$. Like in the first case we have $\dfrac{12\times 10}{4}$ and in the second case we have $\dfrac{12\times 100}{4}$ . Hence we can see that we get the value of 12 × ${{5}^{n}}$ $\dfrac{12({{10}^{n}})}{{{2}^{n}}}$ . Now we can easily calculate the value of $\text{a) 12 }\!\!\times\!\!\text{ 125}$ and $\text{b) 12}\times \text{625}$ as \[{{5}^{3}}\] is 125 and ${{5}^{4}}$ is 625 as the value will be $\dfrac{12{{\left( 10 \right)}^{3}}}{{{2}^{3}}}$ and $\dfrac{12{{\left( 10 \right)}^{4}}}{{{2}^{4}}}$ respectively.

Complete step by step answer:
Now consider the equation 12 × 5 = 12 × $\dfrac{10}{2}$ = $\dfrac{120}{2}$ = 60
We can rewrite the equation as $12\times {{5}^{1}}=\dfrac{12(10)}{2}=\dfrac{12({{10}^{1}})}{{{2}^{1}}}$
Similarly consider the equation 12 × 25 = 12 × $\dfrac{100}{4}$ = $\dfrac{1200}{4}=300$
We can rewrite the equation as $12\times {{5}^{2}}=\dfrac{12(100)}{4}=\dfrac{12({{10}^{2}})}{{{2}^{2}}}$
Hence we can write the general equation of $12\times {{5}^{n}}$ as
$12\times {{5}^{n}}=\dfrac{12({{10}^{n}})}{{{2}^{n}}}.........................(1)$
Now consider 12 × 125
We know that 125 is nothing but cube of 5
Now considering the equation (1) we can write it as $12\times {{5}^{3}}=\dfrac{12({{10}^{3}})}{{{2}^{3}}}=\dfrac{12000}{8}=1500$
or as we know $12\times 125=12\times \dfrac{(1000)}{8}=\dfrac{12000}{8}=1500$
Hence we have $12\times 125=1500..............(1)$
Now consider 12 × 625
We know that 625 is nothing but ${{5}^{4}}$
Hence we have $12\times {{5}^{4}}=\dfrac{12({{10}^{4}})}{{{2}^{4}}}=\dfrac{120000}{16}=7500$ or as we know.$12\times 625=12\times \dfrac{10000}{16}=\dfrac{120000}{16}=7500$

Hence we have $12\times 625=75000$

Note: Here note that while taking example we have ${{2}^{1}}=2$ and ${{2}^{2}}=4$ . Similarly $2(1)=2$ and $2(2)=4$ . Hence do not mistaken the general form $12\times {{5}^{n}}=\dfrac{12({{10}^{n}})}{{{2}^{n}}}$ with $12\times {{5}^{n}}=\dfrac{12({{10}^{n}})}{2n}$