Simplify and leave your answer with positive index
2^n-4×16^1+n
/
8^-n×4^-4n​

2¹⁶ⁿ

Explanation:

2ⁿ¯⁴ × 16¹⁺ⁿ / 8¯ⁿ × 4¯⁴ⁿ

The above expression can be simplified as follow:

2ⁿ¯⁴ × 16¹⁺ⁿ / 8¯ⁿ × 4¯⁴ⁿ

Recall:

16 = 2⁴

8 = 2³

4 = 2²

Thus,

2ⁿ¯⁴ × 16¹⁺ⁿ / 8¯ⁿ × 4¯⁴ⁿ

= 2ⁿ¯⁴ × 2⁴⁽¹⁺ⁿ⁾ / 2³⁽¯ⁿ⁾ × 2²⁽¯⁴ⁿ⁾

= 2ⁿ¯⁴ × 2⁴⁺⁴ⁿ / 2¯³ⁿ × 2¯⁸ⁿ

Recall:

Mᵃ × Mᵇ = Mᵃ⁺ᵇ

2ⁿ¯⁴ × 2⁴⁺⁴ⁿ / 2¯³ⁿ × 2¯⁸ⁿ

= 2⁽ⁿ¯⁴⁾⁺⁽⁴⁺⁴ⁿ⁾ / 2¯³ⁿ⁺⁽¯⁸ⁿ⁾

= 2ⁿ¯⁴⁺⁴⁺⁴ⁿ/ 2¯³ⁿ¯⁸ⁿ

= 2⁵ⁿ / 2¯¹¹ⁿ

Recall:

Mᵃ / Mᵇ = Mᵃ¯ᵇ

2⁵ⁿ / 2¯¹¹ⁿ = 2⁵ⁿ¯⁽¯¹¹ⁿ⁾

2⁵ⁿ / 2¯¹¹ⁿ = 2⁵ⁿ⁺¹¹ⁿ

2⁵ⁿ / 2¯¹¹ⁿ = 2¹⁶ⁿ

Therefore,

2ⁿ¯⁴ × 16¹⁺ⁿ / 8¯ⁿ × 4¯⁴ⁿ = 2¹⁶ⁿ

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