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Question: Simplify \(\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}\)....

Simplify 3−4×10−5×1255−7×6−5\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}.

Explanation

Solution

We have given an expression in exponent and power form and we have to simplify it. Firstly, in the expression, we write 10 and 6 into its factor. 10 has factors 2 and 5, 6 has factors 2 and 3. Then, we apply properties on the factor of the same base. This will reduce the expression in the product of 2, 3, and 5. We apply for the property again and will find the answer.

Complete step-by-step answer:
We have given that 3−4×10−5×1255−7×6−5\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}
We have to simplify it
We know that 1010 can be written as the product of 22 and 55
That is 10=2×510 = 2 \times 5
So, 10−5=(2×5)−5{10^{ - 5}} = {\left( {2 \times 5} \right)^{ - 5}}
Also, we have the property that (a×b)n=an×bn{\left( {a \times b} \right)^n} = {a^n} \times {b^n}
So, (2×5)−5{\left( {2 \times 5} \right)^{ - 5}} can be written as the product of 2−5{2^{ - 5}} and 5−5{5^{ - 5}}
So, 10−5=2−5×5−5{10^{ - 5}} = {2^{ - 5}} \times {5^{ - 5}}
125125 can be written as the product of three 55 that is:
125=5×5×5=53125 = 5 \times 5 \times 5 = {5^3}
66 can be written as the product of 22 and 33 that is:
6=2×36 = 2 \times 3
So, 6−5{6^{ - 5}} can be written as
6−5=(2×3)−5{6^{ - 5}} = {\left( {2 \times 3} \right)^{ - 5}}
=2−5×3−5= {2^{ - 5}} \times {3^{ - 5}}
Now we put all these values in the expression, we get:
3−4×10−5×1255−7×6−5=3−4×10−5×535−7×2−5×3−5\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}} = \dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times {5^3}}}{{{5^{ - 7}} \times {2^{ - 5}} \times {3^{ - 5}}}}
=3−4×2−5×5−5×535−7×2−5×3−5= \dfrac{{{3^{ - 4}} \times {2^{ - 5}} \times {5^{ - 5}} \times {5^3}}}{{{5^{ - 7}} \times {2^{ - 5}} \times {3^{ - 5}}}}
Now, in the numerator we have 5−5×53{5^{ - 5}} \times {5^3}. We have the property that if the base is the same and are in the product then powers can be added.
So, 5−5×53=5−5+3=5−2{5^{ - 5}} \times {5^3} = {5^{ - 5 + 3}} = {5^{ - 2}}
⇒3−4×10−5×1255−7×6−5=2−5×3−4×5−22−5×3−5×5−7=(2−52−5)×(3−43−5)×(5−25−7)\Rightarrow \dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}} = \dfrac{{{2^{ - 5}} \times {3^{ - 4}} \times {5^{ - 2}}}}{{{2^{ - 5}} \times {3^{ - 5}} \times {5^{ - 7}}}} = \left( {\dfrac{{{2^{ - 5}}}}{{{2^{ - 5}}}}} \right) \times \left( {\dfrac{{{3^{ - 4}}}}{{{3^{ - 5}}}}} \right) \times \left( {\dfrac{{{5^{ - 2}}}}{{{5^{ - 7}}}}} \right)
We have the property that if bases are the same and are in the division, then powers can be subtracted.
So, 3−4×10−5×1255−7×6−5=2−5−(−5)×3−4−(−5)×5−2−(−7)\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}} = {2^{ - 5 - \left( { - 5} \right)}} \times {3^{ - 4 - \left( { - 5} \right)}} \times {5^{ - 2 - \left( { - 7} \right)}}
=2−5+5×3−4+5×5−2+7= {2^{ - 5 + 5}} \times {3^{ - 4 + 5}} \times {5^{ - 2 + 7}}
=20×31×55= {2^0} \times {3^1} \times {5^5}
=1×3×5×5×5×5×5= 1 \times 3 \times 5 \times 5 \times 5 \times 5 \times 5
=9375= 9375

So, 3−4×10−5×1255−7×6−5=9375\dfrac{{{3^{ - 4}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}} = 9375

Note: Power denotes the repeated multiplication of the factors and the number which is raised to the base factor is the exponent. This is the main difference between power and exponent.
For example: 32{3^2} is the power where 33 is the base and 22 is the exponent.
Base Number: A base number is a number that is multiplied by itself.