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what is $4\times12$ raised to $4 \times9$ raised to $3 \div27 \times8$ raised to $2 \times 6$ raised to $3$ in exponential form

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Convert between exponential and logarithmic form: all bases

Full solution

Q. what is

4\times12

raised to

4 \times9

raised to

3 \div27 \times8

raised to

2 \times 6

raised to

3

in exponential form

Write Expression: First, let's write down the expression in its current form to understand what we are working with. $\newline$ Expression: (\(4\times $12$ )^ $4$ \times ( $9$ )^ $3$ \div ( $27$ ) \times ( $8$ )^ $2$ \times ( $6$ )^ $3$
Simplify Bases: Now, let's simplify the expression by breaking down the bases into prime factors to make it easier to work with exponents. $\newline$ $4 = 2^2$ $\newline$ $12 = 2^2 \times 3$ $\newline$ $9 = 3^2$ $\newline$ $27 = 3^3$ $\newline$ $8 = 2^3$ $\newline$ $6 = 2 \times 3$
Substitute Prime Factors: Next, we substitute the prime factors into the expression. $\newline$ Expression: $((2^2\times2^2\times3)^4 \times (3^2)^3 \div (3^3) \times (2^3)^2 \times (2\times3)^3)$
Apply Exponents: Now, apply the exponents to the prime factors within the parentheses. $\newline$ Expression: $(2^8\times3^4)^4 \times 3^6 \div 3^3 \times 2^6 \times (2^3\times3^3)$
Distribute Exponents: Next, distribute the exponents to each prime factor. $\newline$ Expression: $2^{(8\times4)}\times3^{(4\times4)} \times 3^6 \div 3^3 \times 2^6 \times 2^9\times3^9$
Simplify Exponents: Now, simplify the exponents by performing the multiplication. $\newline$ Expression: $2^{32}\times3^{16} \times 3^{6} \div 3^{3} \times 2^{6} \times 2^{9}\times3^{9}$
Combine Like Bases: Combine the like bases by adding or subtracting the exponents, as per the laws of exponents. $\newline$ Expression: $2^{32+6+9}\times3^{16+6-3+9}$
Perform Addition/Subtraction: Now, perform the addition and subtraction of the exponents. Expression: $2^{47}\times3^{28}$
Final Simplified Expression: Finally, we have the expression in its simplest exponential form. $\newline$ Expression: $2^{47}\times3^{28}$

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