Simplify the following: $$ \begin{array}{lll} ext { (a) } \frac{2^{2 x}-2^{x}}{2^{x}-1} & ext { (b) } \frac{2^{2 x}-1}{2^{x}+1} & ext { (c) } \frac{9^{x}-9}{3^{x}-3} \ ext { (d) } \frac{16-4^{x}}{2^{x}-4} & \end{array} $$

Question

Simplify the following: $$ \begin{array}{lll}	ext { (a) } \frac{2^{2 x}-2^{x}}{2^{x}-1} & 	ext { (b) } \frac{2^{2 x}-1}{2^{x}+1} & 	ext { (c) } \frac{9^{x}-9}{3^{x}-3} \ 	ext { (d) } \frac{16-4^{x}}{2^{x}-4} & \end{array} $$

UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ (2^(2*x)-2^x)/(2^x-1) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(2^{2x}-2^{x}ight)}{\left(2^{x}-1ight)}$$
- step1: Remove the parentheses:
  $$\frac{2^{2x}-2^{x}}{2^{x}-1}$$
- step2: Rewrite the expression:
  $$\frac{\left(2^{x}ight)^{2}-2^{x}}{2^{x}-1}$$
- step3: Factor the expression:
  $$\frac{2^{x}\left(2^{x}-1ight)}{2^{x}-1}$$
- step4: Reduce the fraction:
  $$2^{x}$$
Calculate or simplify the expression $$ (2^(2*x)-1)/(2^x+1) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(2^{2x}-1ight)}{\left(2^{x}+1ight)}$$
- step1: Remove the parentheses:
  $$\frac{2^{2x}-1}{2^{x}+1}$$
- step2: Rewrite the expression:
  $$\frac{\left(2^{x}ight)^{2}-1}{2^{x}+1}$$
- step3: Calculate:
  $$\frac{\left(2^{x}-1ight)\left(2^{x}+1ight)}{2^{x}+1}$$
- step4: Reduce the fraction:
  $$2^{x}-1$$
Calculate or simplify the expression $$ (9^x-9)/(3^x-3) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(9^{x}-9ight)}{\left(3^{x}-3ight)}$$
- step1: Remove the parentheses:
  $$\frac{9^{x}-9}{3^{x}-3}$$
- step2: Evaluate:
  $$\frac{3\left(3^{2x-1}-3ight)}{3\left(3^{x-1}-1ight)}$$
- step3: Evaluate:
  $$\frac{3^{2x-1}-3}{3^{x-1}-1}$$
- step4: Rewrite the expression:
  $$\frac{\frac{1}{3}\left(3^{x}ight)^{2}-3}{\frac{1}{3}	imes 3^{x}-1}$$
- step5: Rewrite the expression:
  $$\frac{\frac{\left(3^{x}ight)^{2}-9}{3}}{\frac{1}{3}	imes 3^{x}-1}$$
- step6: Rewrite the expression:
  $$\frac{\frac{\left(3^{x}ight)^{2}-9}{3}}{\frac{3^{x}-3}{3}}$$
- step7: Multiply by the reciprocal:
  $$\frac{\left(3^{x}ight)^{2}-9}{3}	imes \frac{3}{3^{x}-3}$$
- step8: Rewrite the expression:
  $$\frac{\left(3^{x}-3ight)\left(3^{x}+3ight)}{3}	imes \frac{3}{3^{x}-3}$$
- step9: Reduce the fraction:
  $$\left(3^{x}+3ight)	imes 1$$
- step10: Multiply the terms:
  $$3^{x}+3$$
Calculate or simplify the expression $$ (16-4^x)/(2^x-4) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(16-4^{x}ight)}{\left(2^{x}-4ight)}$$
- step1: Remove the parentheses:
  $$\frac{16-4^{x}}{2^{x}-4}$$
- step2: Evaluate:
  $$\frac{4\left(2^{2}-2^{2x-2}ight)}{4\left(2^{x-2}-1ight)}$$
- step3: Evaluate:
  $$\frac{2^{2}-2^{2x-2}}{2^{x-2}-1}$$
- step4: Calculate:
  $$\frac{4-2^{2x-2}}{2^{x-2}-1}$$
- step5: Rewrite the expression:
  $$\frac{4-\frac{1}{4}\left(2^{x}ight)^{2}}{\frac{1}{4}	imes 2^{x}-1}$$
- step6: Rewrite the expression:
  $$\frac{\frac{16-\left(2^{x}ight)^{2}}{4}}{\frac{1}{4}	imes 2^{x}-1}$$
- step7: Rewrite the expression:
  $$\frac{\frac{16-\left(2^{x}ight)^{2}}{4}}{\frac{2^{x}-4}{4}}$$
- step8: Multiply by the reciprocal:
  $$\frac{16-\left(2^{x}ight)^{2}}{4}	imes \frac{4}{2^{x}-4}$$
- step9: Rewrite the expression:
  $$\frac{\left(2^{x}-4ight)\left(-4-2^{x}ight)}{4}	imes \frac{4}{2^{x}-4}$$
- step10: Reduce the fraction:
  $$\left(-4-2^{x}ight)	imes 1$$
- step11: Multiply the terms:
  $$-\left(4+2^{x}ight)$$
- step12: Calculate:
  $$-4-2^{x}$$
The simplified expressions are:
(a) $$2^{x}$$
(b) $$2^{x}-1$$
(c) $$3^{x}+3$$
(d) $$-4-2^{x}$$
Simplify the expressions as follows: (a) $$2^{x}$$ (b) $$2^{x}-1$$ (c) $$3^{x}+3$$ (d) $$-4-2^{x}$$

Simplify the following: \( \begin{array}{lll}\text { (a) } \frac{2^{2 x}-2^{x}}{2^{x}-1} & \text { (b) } \frac{2^{2 x}-1}{2^{x}+1} & \text { (c) } \frac{9^{x}-9}{3^{x}-3} \\ \text { (d) } \frac{16-4^{x}}{2^{x}-4} & \end{array} \)

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