Simplify (37)8\left(3^{7}\right)^{8}(3 7 ) 8 . Write your answer in exponential form. (Example: 525^25 2 not 252525 )

Answer 1

To simplify the expression $(37)8{\left(3^7\right)}^8(37)8$, let's break it down step by step.

1. Understanding Each Component:

   • The term $(37)8$ can be interpreted as ${37}^8$.
   • The term $(3^7{)}^8$ can be simplified using the power of a power rule, which states that $(a^m{)}^n=a^{m\cdot n}$. Therefore, $(3^7{)}^8=3^{7\cdot 8}=3^{56}$.
   • The term $(37)8$ can also be interpreted as being multiplied by ${37}^8$, i.e., $3^8\cdot 7^8$.

2. Combining Terms:

   • Now we can combine everything: $${37}^8\cdot 3^{56}\cdot 3^8\cdot 7^8$$
   • We can combine the bases for $3$: $$3^{56}\cdot 3^8=3^{56+8}=3^{64}$$

3. Revisiting the Expression:

   • Now rewriting our expression gives: $${37}^8\cdot 3^{64}\cdot 7^8$$

4. Final Form:

   • The expression can be rearranged to group the powers: $$3^{64}\cdot {37}^8\cdot 7^8$$
   • We can represent ${37}^8\cdot 7^8$ together using the property of exponentiation $a^m{b}^m=(ab{)}^m$: $${37}^8\cdot 7^8=(37\cdot 7{)}^8={259}^8\text{ (since }37\times 7=259)$$

5. Resulting Exponential Form:

   • Finally, we combine everything: $$3^{64}\cdot {259}^8$$

Thus, in exponential form, the simplified expression is:

$$3^{64}\cdot {259}^8$$