Solve the following equations for $$ x $$. $$ \begin{array}{ll}17 .) 5^{2 x}=5^{2} & 18 \ ext { 19. }(2.5)^{2 x+1}=(2.5)^{5} & 20 \ ext { 21. } 10^{1-x}=100 & 22 \ ext { 23. } 3(2.7)^{5 x}=8.1 & 24 \ ext { 25. }\left(2^{x+1} \cdot 2^{-3} ight)^{2}=2 & 26\end{array} $$

Question

Solve the following equations for $$ x $$. $$ \begin{array}{ll}17 .) 5^{2 x}=5^{2} & 18 \ 	ext { 19. }(2.5)^{2 x+1}=(2.5)^{5} & 20 \ 	ext { 21. } 10^{1-x}=100 & 22 \ 	ext { 23. } 3(2.7)^{5 x}=8.1 & 24 \ 	ext { 25. }\left(2^{x+1} \cdot 2^{-3}ight)^{2}=2 & 26\end{array} $$

UpStudy AI · Accepted Answer

Solve the equation $$ 5^{2x}=5^{2} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$5^{2x}=5^{2}$$
- step1: Set the exponents equal:
  $$2x=2$$
- step2: Divide both sides:
  $$\frac{2x}{2}=\frac{2}{2}$$
- step3: Divide the numbers:
  $$x=1$$
Solve the equation $$ (2.5)^{2x+1}=(2.5)^{5} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2.5^{2x+1}=2.5^{5}$$
- step1: Set the exponents equal:
  $$2x+1=5$$
- step2: Move the constant to the right side:
  $$2x=5-1$$
- step3: Subtract the numbers:
  $$2x=4$$
- step4: Divide both sides:
  $$\frac{2x}{2}=\frac{4}{2}$$
- step5: Divide the numbers:
  $$x=2$$
Solve the equation $$ 10^{1-x}=100 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$10^{1-x}=100$$
- step1: Rewrite in exponential form:
  $$10^{1-x}=10^{2}$$
- step2: Set the exponents equal:
  $$1-x=2$$
- step3: Move the constant to the right side:
  $$-x=2-1$$
- step4: Subtract the numbers:
  $$-x=1$$
- step5: Change the signs:
  $$x=-1$$
Solve the equation $$ 3(2.7)^{5x}=8.1 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$3	imes 2.7^{5x}=8.1$$
- step1: Convert the expressions:
  $$3\left(\frac{27}{10}ight)^{5x}=8.1$$
- step2: Multiply by the reciprocal:
  $$3\left(\frac{27}{10}ight)^{5x}	imes \frac{1}{3}=8.1	imes \frac{1}{3}$$
- step3: Multiply:
  $$\left(\frac{27}{10}ight)^{5x}=\frac{27}{10}$$
- step4: Set the exponents equal:
  $$5x=1$$
- step5: Divide both sides:
  $$\frac{5x}{5}=\frac{1}{5}$$
- step6: Divide the numbers:
  $$x=\frac{1}{5}$$
Solve the equation $$ \left(2^{x+1} \cdot 2^{-3}ight)^{2}=2 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\left(2^{x+1}	imes 2^{-3}ight)^{2}=2$$
- step1: Simplify:
  $$2^{2\left(x-2ight)}=2$$
- step2: Set the exponents equal:
  $$2\left(x-2ight)=1$$
- step3: Divide both sides:
  $$\frac{2\left(x-2ight)}{2}=\frac{1}{2}$$
- step4: Divide the numbers:
  $$x-2=\frac{1}{2}$$
- step5: Move the constant to the right side:
  $$x=\frac{1}{2}+2$$
- step6: Add the numbers:
  $$x=\frac{5}{2}$$
The solutions to the given equations are:

1. $$5^{2x}=5^{2}$$ has the solution $$x=1$$.

2. $$(2.5)^{2x+1}=(2.5)^{5}$$ has the solution $$x=2$$.

3. $$10^{1-x}=100$$ has the solution $$x=-1$$.

4. $$3(2.7)^{5x}=8.1$$ has the solution $$x=\frac{1}{5}$$ or $$x=0.2$$.

5. $$\left(2^{x+1} \cdot 2^{-3}ight)^{2}=2$$ has the solution $$x=\frac{5}{2}$$ or $$x=2.5$$.
The solutions to the equations are: 1. $$x = 1$$ 2. $$x = 2$$ 3. $$x = -1$$ 4. $$x = 0.2$$ or $$x = 0.2$$ 5. $$x = 2.5$$

Solve the following equations for \( x \). \( \begin{array}{ll}17 .) 5^{2 x}=5^{2} & 18 \\ \text { 19. }(2.5)^{2 x+1}=(2.5)^{5} & 20 \\ \text { 21. } 10^{1-x}=100 & 22 \\ \text { 23. } 3(2.7)^{5 x}=8.1 & 24 \\ \text { 25. }\left(2^{x+1} \cdot 2^{-3}\right)^{2}=2 & 26\end{array} \)

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