Question

Simplify: (12^4 × 9^3 × 4)/(6^3 × 8^2 × 27) = ?​

Answer 1

Answer:

★ How to do:-

Here, we are given with the fraction containing of exponential numbers. One of the number in denominator is in the whole number form which should be converted into exponential form. The composite numbers should be converted into the powers of prime numbers by using the method if prime factorization as a rough work. The most useful formulas are given below known as six laws of exponents. So, let's solve!!

$\:$

➤ Solution :-

${\tt \leadsto \dfrac{{12}^{4} \times {9}^{3} \times 4}{{6}^{3} \times {8}^{2} \times 27}}$

${\tt \leadsto \dfrac{{12}^{4} \times {9}^{3} \times {2}^{2}}{{6}^{3} \times {8}^{2} \times {3}^{3}}}$

${\tt \leadsto \dfrac{({3}^{1} \times {2}^{2}{)}^{4} \times ({3}^{2}{)}^{3} \times {2}^{2}}{{(2 \times 3)}^{3} \times {({2}^{3})}^{2} \times {3}^{3}}}$

${\tt \leadsto \dfrac{{3}^{1 \times 4} \times {2}^{2 \times 4} \times {3}^{2 \times 3} \times {2}^{2}}{{2}^{3} \times {{3}^{3}} \times {2}^{3 \times 2} \times {3}^{3}}}$

${\tt \leadsto \dfrac{{3}^{4} \times {2}^{8} \times {3}^{6} \times {2}^{2}}{{2}^{3} \times {{3}^{3}} \times {2}^{6} \times {3}^{3}}}$

${\tt \leadsto \dfrac{{3}^{4 + 6} \times {2}^{8 + 2}}{{2}^{3 + 6} \times {{3}^{3 + 3}}}}$

${\tt \leadsto \dfrac{{3}^{10} \times {2}^{10}}{{2}^{9} \times {{3}^{6}}}}$

${\tt \leadsto {3}^{10 - 6} \times {2}^{10 - 9}}$

${\tt \leadsto {3}^{4} \times {2}^{1}}$

${\tt \leadsto 3 \times 3 \times 3 \times 3 \times 2}$

${\tt \leadsto 81 \times 2 = \boxed{\tt 162}}$

$\Huge\therefore$ The answer is 162.

━━━━━━━━━━━━━━━━━━━━━━

$\dashrightarrow$ Some related equations :-

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

${\sf \longrightarrow {a}^{m} \div {a}^{n} = {a}^{m - n}}$

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

${\sf \longrightarrow {a}^{ - n} = \dfrac{1}{ {a}^{n}}}$

${\sf \longrightarrow {a}^{0} = 1}$

Answer 2

Answer:

➤ Solution :-

{\tt \leadsto \dfrac{{12}^{4} \times {9}^{3} \times 4}{{6}^{3} \times {8}^{2} \times 27}}⇝

6

3

×8

2

×27

12

4

×9

3

×4

{\tt \leadsto \dfrac{{12}^{4} \times {9}^{3} \times {2}^{2}}{{6}^{3} \times {8}^{2} \times {3}^{3}}}⇝

6

3

×8

2

×3

3

12

4

×9

3

×2

2

{\tt \leadsto \dfrac{({3}^{1} \times {2}^{2}{)}^{4} \times ({3}^{2}{)}^{3} \times {2}^{2}}{{(2 \times 3)}^{3} \times {({2}^{3})}^{2} \times {3}^{3}}}⇝

(2×3)

3

×(2

3

)

2

×3

3

(3

1

×2

2

)

4

×(3

2

)

3

×2

2

{\tt \leadsto \dfrac{{3}^{1 \times 4} \times {2}^{2 \times 4} \times {3}^{2 \times 3} \times {2}^{2}}{{2}^{3} \times {{3}^{3}} \times {2}^{3 \times 2} \times {3}^{3}}}⇝

2

3

×3

3

×2

3×2

×3

3

3

1×4

×2

2×4

×3

2×3

×2

2

{\tt \leadsto \dfrac{{3}^{4} \times {2}^{8} \times {3}^{6} \times {2}^{2}}{{2}^{3} \times {{3}^{3}} \times {2}^{6} \times {3}^{3}}}⇝

2

3

×3

3

×2

6

×3

3

3

4

×2

8

×3

6

×2

2

{\tt \leadsto \dfrac{{3}^{4 + 6} \times {2}^{8 + 2}}{{2}^{3 + 6} \times {{3}^{3 + 3}}}}⇝

2

3+6

×3

3+3

3

4+6

×2

8+2

{\tt \leadsto \dfrac{{3}^{10} \times {2}^{10}}{{2}^{9} \times {{3}^{6}}}}⇝

2

9

×3

6

3

10

×2

10

{\tt \leadsto {3}^{10 - 6} \times {2}^{10 - 9}}⇝3

10−6

×2

10−9

{\tt \leadsto {3}^{4} \times {2}^{1}}⇝3

4

×2

1

{\tt \leadsto 3 \times 3 \times 3 \times 3 \times 2}⇝3×3×3×3×2

{\tt \leadsto 81 \times 2 = \boxed{\tt 162}}⇝81×2=

162

\Huge\therefore∴ The answer is 162.

━━━━━━━━━━━━━━━━━━━━━━

\dashrightarrow⇢ Some related equations :-

{\sf \longrightarrow {a}^{m} \times {a}^{n} = {a}^{m + n}}⟶a

m

×a

n

=a

m+n

{\sf \longrightarrow {a}^{m} \div {a}^{n} = {a}^{m - n}}⟶a

m

÷a

n

=a

m−n

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

m

)

n

=a

m×n

{\sf \longrightarrow {a}^{ - n} = \dfrac{1}{ {a}^{n}}}⟶a

−n

=

a

n

1

{\sf \longrightarrow {a}^{0} = 1}⟶a

0

=1