Question

[5^7/5^2] into 5^3= ???

Answer 1

We have been asked to find the value of $\sf{(\dfrac{{(5)}^{7}}{{(5)}^{2}} \times {(5)}^{3})}$ .

If we observe carefully ; the bases are the same while the powers are different. We can find the solution simply by operating the powers and keeping the bases constant.

$\longrightarrow{\sf{(\dfrac{{(5)}^{7}}{{(5)}^{2}} \times {(5)}^{3})}}$

According to the identity which states :

$\longrightarrow{\sf{(\dfrac{{(a)}^{m}}{{(a)}^{n}} = {(a)}^{m - n})}}$

So following the same identity we can solve this as :

$\longrightarrow{\sf{(\dfrac{{(5)}^{7}}{{(5)}^{2}} = {(5)}^{7 - 2})}}$

$\longrightarrow{\sf{(\dfrac{{(5)}^{7}}{{(5)}^{2}} = {(5)}^{5})}}$

According to another identity :

$\longrightarrow{\sf{({(a)}^{m} \times {(a)}^{n} = {(a)}^{m+n})}}$

Following the same we solve it as :

$\longrightarrow{\sf{({(5)}^{5} \times {(5)}^{3} = {(5)}^{5+3})}}$

$\longrightarrow{\sf{({(5)}^{5} \times {(5)}^{3} = {(5)}^{8})}}$

So the value is :

$\large{\boxed{\implies{\sf{{(5)}^{8}}}}}$

Verification :

$\longrightarrow{\sf{{(5)}^{7} \div {(5)}^{2} \times {(5)}^{3} = {(5)}^{8} }}$

$\longrightarrow{\sf{78125 \div 25 \times 125 = 390625}}$

$\longrightarrow{\sf{\cancel{\dfrac{78125}{25}} \times 125 = 390625}}$

$\longrightarrow{\sf{3125 \times 125 = 390625}}$

$\longrightarrow{\sf{390625 = 390625}}$

LHS = RHS