Simplify the following: $$ \begin{array}{lll} ext { (a) } \frac{10^{x} \cdot 25^{x+1}}{5^{x} \cdot 50^{x-1}} & ext { (b) } \frac{6^{n+2} imes 10^{n-2}}{4^{n} imes 15^{n-2}} & ext { (c) } \frac{6^{x} \cdot 9^{x+1} \cdot 2}{27^{x+1} \cdot 2^{x-1}} \ ext { (d) } \frac{8^{n} \cdot 6^{n-3} \cdot 9^{1-n}}{16^{n-1} \cdot 3^{-n}} & ext { (e) } \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} & ext { (f) } \frac{9^{x}+3^{x+1}}{18^{x} \cdot 2^{1-x}} \ ext { (g) } \frac{3 \cdot 2^{x}-2^{x-1}}{2^{x}+2^{x+2}} & ext { (h) } \frac{4^{x}+2^{2 x-1}}{2^{2 x-1}} & ext { (i) } \frac{2.3^{x+2}+3^{x-3}}{5 \cdot 3^{x-2}}\end{array} $$

Question

Simplify the following: $$ \begin{array}{lll}	ext { (a) } \frac{10^{x} \cdot 25^{x+1}}{5^{x} \cdot 50^{x-1}} & 	ext { (b) } \frac{6^{n+2} 	imes 10^{n-2}}{4^{n} 	imes 15^{n-2}} & 	ext { (c) } \frac{6^{x} \cdot 9^{x+1} \cdot 2}{27^{x+1} \cdot 2^{x-1}} \ 	ext { (d) } \frac{8^{n} \cdot 6^{n-3} \cdot 9^{1-n}}{16^{n-1} \cdot 3^{-n}} & 	ext { (e) } \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} & 	ext { (f) } \frac{9^{x}+3^{x+1}}{18^{x} \cdot 2^{1-x}} \ 	ext { (g) } \frac{3 \cdot 2^{x}-2^{x-1}}{2^{x}+2^{x+2}} & 	ext { (h) } \frac{4^{x}+2^{2 x-1}}{2^{2 x-1}} & 	ext { (i) } \frac{2.3^{x+2}+3^{x-3}}{5 \cdot 3^{x-2}}\end{array} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(4^{x}+2^{2x-1}\right)}{2^{2x-1}}$$
- step1: Remove the parentheses:
  $$\frac{4^{x}+2^{2x-1}}{2^{2x-1}}$$
- step2: Add the terms:
  $$\frac{3\times 2^{2x-1}}{2^{2x-1}}$$
- step3: Rewrite the expression:
  $$3\times 1$$
- step4: Multiply:
  $$3$$
Calculate or simplify the expression $$ (10^x * 25^(x+1)) / (5^x * 50^(x-1)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(10^{x}\times 25^{x+1}\right)}{\left(5^{x}\times 50^{x-1}\right)}$$
- step1: Remove the parentheses:
  $$\frac{10^{x}\times 25^{x+1}}{5^{x}\times 50^{x-1}}$$
- step2: Factor the expression:
  $$\frac{5^{x}\times 2^{x}\times 25^{x+1}}{5^{x}\times 50^{x-1}}$$
- step3: Reduce the fraction:
  $$\frac{2^{x}\times 25^{x+1}}{50^{x-1}}$$
- step4: Factor the expression:
  $$\frac{2^{x}\times 25^{x+1}}{2^{x-1}\times 25^{x-1}}$$
- step5: Reduce the fraction:
  $$2\times 25^{2}$$
- step6: Evaluate the power:
  $$2\times 625$$
- step7: Multiply:
  $$1250$$
Calculate or simplify the expression $$ (2 * 3^(x+2) + 3^(x-3)) / (5 * 3^(x-2)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(2\times 3^{x+2}+3^{x-3}\right)}{\left(5\times 3^{x-2}\right)}$$
- step1: Remove the parentheses:
  $$\frac{2\times 3^{x+2}+3^{x-3}}{5\times 3^{x-2}}$$
- step2: Add the terms:
  $$\frac{487\times 3^{x-3}}{5\times 3^{x-2}}$$
- step3: Reduce the fraction:
  $$\frac{487}{5\times 3}$$
- step4: Calculate:
  $$\frac{487}{15}$$
Calculate or simplify the expression $$ (2^(x+2) - 2^(x+3)) / (2^(x+1) - 2^(x+2)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(2^{x+2}-2^{x+3}\right)}{\left(2^{x+1}-2^{x+2}\right)}$$
- step1: Remove the parentheses:
  $$\frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}}$$
- step2: Subtract the terms:
  $$\frac{-2^{x+2}}{2^{x+1}-2^{x+2}}$$
- step3: Subtract the terms:
  $$\frac{-2^{x+2}}{-2^{x+1}}$$
- step4: Calculate:
  $$2^{x+2-\left(x+1\right)}$$
- step5: Calculate:
  $$2$$
Calculate or simplify the expression $$ (6^x * 9^(x+1) * 2) / (27^(x+1) * 2^(x-1)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(6^{x}\times 9^{x+1}\times 2\right)}{\left(27^{x+1}\times 2^{x-1}\right)}$$
- step1: Remove the parentheses:
  $$\frac{6^{x}\times 9^{x+1}\times 2}{27^{x+1}\times 2^{x-1}}$$
- step2: Multiply by $$a^{-n}:$$
  $$\frac{6^{x}\times 9^{x+1}\times 2\times 2^{-\left(x-1\right)}}{27^{x+1}}$$
- step3: Calculate:
  $$\frac{6^{x}\times 9^{x+1}\times 2\times 2^{-x+1}}{27^{x+1}}$$
- step4: Multiply:
  $$\frac{6^{x}\times 9^{x+1}\times 2^{2-x}}{27^{x+1}}$$
- step5: Factor the expression:
  $$\frac{3^{x}\times 2^{x}\times 9^{x+1}\times 2^{2-x}}{3^{3x+3}}$$
- step6: Reduce the fraction:
  $$\frac{2^{x}\times 9^{x+1}\times 2^{2-x}}{3^{2x+3}}$$
- step7: Factor the expression:
  $$\frac{2^{x}\times 3^{2x+2}\times 2^{2-x}}{3^{2x+3}}$$
- step8: Reduce the fraction:
  $$\frac{2^{x}\times 2^{2-x}}{3}$$
- step9: Multiply the terms:
  $$\frac{2^{2}}{3}$$
- step10: Evaluate the power:
  $$\frac{4}{3}$$
Calculate or simplify the expression $$ (6^(n+2) * 10^(n-2)) / (4^n * 15^(n-2)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(6^{n+2}\times 10^{n-2}\right)}{\left(4^{n}\times 15^{n-2}\right)}$$
- step1: Remove the parentheses:
  $$\frac{6^{n+2}\times 10^{n-2}}{4^{n}\times 15^{n-2}}$$
- step2: Factor the expression:
  $$\frac{2^{n+2}\times 3^{n+2}\times 10^{n-2}}{2^{2n}\times 15^{n-2}}$$
- step3: Reduce the fraction:
  $$\frac{3^{n+2}\times 10^{n-2}}{2^{n-2}\times 15^{n-2}}$$
- step4: Factor the expression:
  $$\frac{3^{n+2}\times 10^{n-2}}{2^{n-2}\times 3^{n-2}\times 5^{n-2}}$$
- step5: Reduce the fraction:
  $$\frac{3^{4}\times 10^{n-2}}{2^{n-2}\times 5^{n-2}}$$
- step6: Factor the expression:
  $$\frac{3^{4}\times 2^{n-2}\times 5^{n-2}}{2^{n-2}\times 5^{n-2}}$$
- step7: Reduce the fraction:
  $$3^{4}$$
- step8: Evaluate the power:
  $$81$$
Calculate or simplify the expression $$ (3 * 2^x - 2^(x-1)) / (2^x + 2^(x+2)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(3\times 2^{x}-2^{x-1}\right)}{\left(2^{x}+2^{x+2}\right)}$$
- step1: Remove the parentheses:
  $$\frac{3\times 2^{x}-2^{x-1}}{2^{x}+2^{x+2}}$$
- step2: Add the terms:
  $$\frac{3\times 2^{x}-2^{x-1}}{5\times 2^{x}}$$
- step3: Factor the expression:
  $$\frac{\left(3-2^{-1}\right)\times 2^{x}}{5\times 2^{x}}$$
- step4: Reduce the fraction:
  $$\frac{3-2^{-1}}{5}$$
- step5: Calculate:
  $$\frac{\frac{5}{2}}{5}$$
- step6: Multiply by the reciprocal:
  $$\frac{5}{2}\times \frac{1}{5}$$
- step7: Reduce the numbers:
  $$\frac{1}{2}\times 1$$
- step8: Multiply:
  $$\frac{1}{2}$$
Calculate or simplify the expression $$ (9^x + 3^(x+1)) / (18^x * 2^(1-x)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(9^{x}+3^{x+1}\right)}{\left(18^{x}\times 2^{1-x}\right)}$$
- step1: Remove the parentheses:
  $$\frac{9^{x}+3^{x+1}}{18^{x}\times 2^{1-x}}$$
- step2: Multiply the terms:
  $$\frac{9^{x}+3^{x+1}}{2\times 9^{x}}$$
- step3: Factor the expression:
  $$\frac{3^{x}\left(3^{x}+3\right)}{2\times 3^{2x}}$$
- step4: Reduce the fraction:
  $$\frac{3^{x}+3}{2\times 3^{x}}$$
- step5: Factor the expression:
  $$\frac{3\left(3^{x-1}+1\right)}{2\times 3^{x}}$$
- step6: Divide the numbers:
  $$\frac{3^{1-x}\left(3^{x-1}+1\right)}{2}$$
- step7: Calculate:
  $$\frac{1+3^{1-x}}{2}$$
Calculate or simplify the expression $$ (8^n * 6^(n-3) * 9^(1-n)) / (16^(n-1) * 3^(-n)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(8^{n}\times 6^{n-3}\times 9^{1-n}\right)}{\left(16^{n-1}\times 3^{-n}\right)}$$
- step1: Remove the parentheses:
  $$\frac{8^{n}\times 6^{n-3}\times 9^{1-n}}{16^{n-1}\times 3^{-n}}$$
- step2: Factor the expression:
  $$\frac{2^{3n}\times 6^{n-3}\times 9^{1-n}}{2^{4n-4}\times 3^{-n}}$$
- step3: Reduce the fraction:
  $$\frac{6^{n-3}\times 9^{1-n}}{2^{n-4}\times 3^{-n}}$$
- step4: Factor the expression:
  $$\frac{2^{n-3}\times 3^{n-3}\times 9^{1-n}}{2^{n-4}\times 3^{-n}}$$
- step5: Reduce the fraction:
  $$2\times 3^{2n-3}\times 9^{1-n}$$
- step6: Transform the expression:
  $$2\times 3^{2n-3}\times 3^{2\left(1-n\right)}$$
- step7: Multiply the terms:
  $$2\times 3^{2n-3+2\left(1-n\right)}$$
- step8: Calculate:
  $$2\times 3^{-1}$$
- step9: Use the properties of exponents:
  $$2\times \frac{1}{3}$$
- step10: Multiply:
  $$\frac{2}{3}$$
Here are the simplified results for each expression:

(a) 
$$
\frac{10^{x} \cdot 25^{x+1}}{5^{x} \cdot 50^{x-1}} = 1250
$$

(b) 
$$
\frac{6^{n+2} \times 10^{n-2}}{4^{n} \times 15^{n-2}} = 81
$$

(c) 
$$
\frac{6^{x} \cdot 9^{x+1} \cdot 2}{27^{x+1} \cdot 2^{x-1}} = \frac{4}{3} \quad \text{(or } 1\frac{1}{3} \text{ or } 1.\dot{3} \text{)}
$$

(d) 
$$
\frac{8^{n} \cdot 6^{n-3} \cdot 9^{1-n}}{16^{n-1} \cdot 3^{-n}} = \frac{2}{3} \quad \text{(or } 0.\dot{6} \text{)}
$$

(e) 
$$
\frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} = 2
$$

(f) 
$$
\frac{9^{x}+3^{x+1}}{18^{x} \cdot 2^{1-x}} = \frac{1+3^{1-x}}{2}
$$

(g) 
$$
\frac{3 \cdot 2^{x}-2^{x-1}}{2^{x}+2^{x+2}} = \frac{1}{2} \quad \text{(or } 0.5 \text{)}
$$

(h) 
$$
\frac{4^{x}+2^{2 x-1}}{2^{2 x-1}} = 3
$$

(i) 
$$
\frac{2 \cdot 3^{x+2}+3^{x-3}}{5 \cdot 3^{x-2}} = \frac{487}{15}
$$

If you need further assistance or explanations for any specific part, feel free to ask!
Here are the simplified results for each expression: (a) $$ 1250 $$ (b) $$ 81 $$ (c) $$ \frac{4}{3} $$ (or $$ 1\frac{1}{3} $$ or $$ 1.\dot{3} $$) (d) $$ \frac{2}{3} $$ (or $$ 0.\dot{6} $$) (e) $$ 2 $$ (f) $$ \frac{1+3^{1-x}}{2} $$ (g) $$ \frac{1}{2} $$ (or $$ 0.5 $$) (h) $$ 3 $$ (i) $$ \frac{487}{15} $$

Simplify the following:

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