Answer
1.1.1: \( \frac{100 \times 193^{x} - 93^{x+2}}{100000 \times 183^{x-3} + 10 \times 393^{x+1}} \)
1.1.2: \( -\frac{1}{5} \) or \( -0.2 \)
Solution
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(\sqrt{48x^{6}}-\left(125x^{9}\right)^{\frac{1}{3}}\times 3^{\frac{1}{2}}\right)}{\sqrt{75x^{6}}}\)
- step1: Remove the parentheses:
\(\frac{\sqrt{48x^{6}}-\left(125x^{9}\right)^{\frac{1}{3}}\times 3^{\frac{1}{2}}}{\sqrt{75x^{6}}}\)
- step2: Simplify the root:
\(\frac{4\sqrt{3}\times x^{3}-\left(125x^{9}\right)^{\frac{1}{3}}\times 3^{\frac{1}{2}}}{\sqrt{75x^{6}}}\)
- step3: Multiply the terms:
\(\frac{4\sqrt{3}\times x^{3}-5\times 3^{\frac{1}{2}}x^{3}}{\sqrt{75x^{6}}}\)
- step4: Simplify the root:
\(\frac{4\sqrt{3}\times x^{3}-5\times 3^{\frac{1}{2}}x^{3}}{5\sqrt{3}\times x^{3}}\)
- step5: Subtract the terms:
\(\frac{-\sqrt{3}\times x^{3}}{5\sqrt{3}\times x^{3}}\)
- step6: Reduce the fraction:
\(\frac{-\sqrt{3}}{5\sqrt{3}}\)
- step7: Reduce the fraction:
\(\frac{-1}{5}\)
- step8: Rewrite the fraction:
\(-\frac{1}{5}\)
Calculate or simplify the expression \( (19.3^x - 9.3^(x+2)) / (18.3^(x-3) + 39.3^(1+x)) \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(19.3^{x}-9.3^{x+2}\right)}{\left(18.3^{x-3}+39.3^{1+x}\right)}\)
- step1: Remove the parentheses:
\(\frac{19.3^{x}-9.3^{x+2}}{18.3^{x-3}+39.3^{1+x}}\)
- step2: Convert the expressions:
\(\frac{\left(\frac{193}{10}\right)^{x}-9.3^{x+2}}{18.3^{x-3}+39.3^{1+x}}\)
- step3: Convert the expressions:
\(\frac{\left(\frac{193}{10}\right)^{x}-\left(\frac{93}{10}\right)^{x+2}}{18.3^{x-3}+39.3^{1+x}}\)
- step4: Convert the expressions:
\(\frac{\left(\frac{193}{10}\right)^{x}-\left(\frac{93}{10}\right)^{x+2}}{\left(\frac{183}{10}\right)^{x-3}+39.3^{1+x}}\)
- step5: Convert the expressions:
\(\frac{\left(\frac{193}{10}\right)^{x}-\left(\frac{93}{10}\right)^{x+2}}{\left(\frac{183}{10}\right)^{x-3}+\left(\frac{393}{10}\right)^{1+x}}\)
- step6: Rewrite the expression:
\(\frac{\left(\frac{193}{10}\right)^{x}-\frac{8649}{100}\left(\frac{93}{10}\right)^{x}}{\frac{1000}{183^{3}}\times \left(\frac{183}{10}\right)^{x}+\frac{393}{10}\left(\frac{393}{10}\right)^{x}}\)
- step7: Rewrite the expression:
\(\frac{\frac{100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}}{100}}{\frac{1000}{183^{3}}\times \left(\frac{183}{10}\right)^{x}+\frac{393}{10}\left(\frac{393}{10}\right)^{x}}\)
- step8: Rewrite the expression:
\(\frac{\frac{100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}}{100}}{\frac{10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}}{10\times 183^{3}}}\)
- step9: Multiply by the reciprocal:
\(\frac{100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}}{100}\times \frac{10\times 183^{3}}{10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}}\)
- step10: Reduce the fraction:
\(\frac{100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}}{10}\times \frac{183^{3}}{10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}}\)
- step11: Multiply the terms:
\(\frac{\left(100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}\right)\times 183^{3}}{10\left(10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}\right)}\)
- step12: Multiply the terms:
\(\frac{183^{3}\left(100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}\right)}{10\left(10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}\right)}\)
- step13: Calculate:
\(\frac{100\times 183^{3}\left(\frac{193}{10}\right)^{x}-8649\times 183^{3}\left(\frac{93}{10}\right)^{x}}{10\left(10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}\right)}\)
- step14: Calculate:
\(\frac{100\times 183^{3}\left(\frac{193}{10}\right)^{x}-8649\times 183^{3}\left(\frac{93}{10}\right)^{x}}{100000\left(\frac{183}{10}\right)^{x}+3930\times 183^{3}\left(\frac{393}{10}\right)^{x}}\)
- step15: Expand the expression:
\(\frac{\frac{100\times 183^{3}\times 193^{x}-183^{3}\times 93^{x+2}}{10^{x}}}{10^{1-x}\left(10000\times 183^{x}+61^{3}\times 3^{4+x}\times 131^{x+1}\right)}\)
- step16: Multiply by the reciprocal:
\(\frac{100\times 183^{3}\times 193^{x}-183^{3}\times 93^{x+2}}{10^{x}}\times \frac{1}{10^{1-x}\left(10000\times 183^{x}+61^{3}\times 3^{4+x}\times 131^{x+1}\right)}\)
- step17: Rewrite the expression:
\(\frac{27\left(100\times 61^{3}\times 193^{x}-61^{3}\times 93^{x+2}\right)}{10^{x}}\times \frac{1}{10^{1-x}\times 27\left(10000\times 3^{x-3}\times 61^{x}+61^{3}\times 393^{x+1}\right)}\)
- step18: Reduce the fraction:
\(\frac{100\times 61^{3}\times 193^{x}-61^{3}\times 93^{x+2}}{10^{x}}\times \frac{1}{10^{1-x}\left(10000\times 3^{x-3}\times 61^{x}+61^{3}\times 393^{x+1}\right)}\)
- step19: Rewrite the expression:
\(\frac{61^{3}\left(100\times 193^{x}-93^{x+2}\right)}{10^{x}}\times \frac{1}{10^{1-x}\times 61^{3}\left(10000\times 183^{x-3}+393^{x+1}\right)}\)
- step20: Reduce the fraction:
\(\frac{100\times 193^{x}-93^{x+2}}{10^{x}}\times \frac{1}{10^{1-x}\left(10000\times 183^{x-3}+393^{x+1}\right)}\)
- step21: Multiply the terms:
\(\frac{100\times 193^{x}-93^{x+2}}{10^{x}\times 10^{1-x}\left(10000\times 183^{x-3}+393^{x+1}\right)}\)
- step22: Multiply the terms:
\(\frac{100\times 193^{x}-93^{x+2}}{100000\times 183^{x-3}+10\times 393^{x+1}}\)
Let's simplify the given expressions step by step.
### 1.1.1 Simplifying \( \frac{19.3^{x}-9.3^{x+2}}{18.3^{x-3}+39.3^{1+x}} \)
1. **Factor out common terms in the numerator:**
\[
19.3^{x} - 9.3^{x+2} = 19.3^{x} - 9 \cdot 3^{2} \cdot 3^{x} = 19.3^{x} - 9 \cdot 9 \cdot 3^{x} = 19.3^{x} - 9 \cdot 9 \cdot 3^{x}
\]
This can be factored as:
\[
3^{x}(19 - 9 \cdot 9) = 3^{x}(19 - 81) = 3^{x}(-62)
\]
2. **Factor out common terms in the denominator:**
\[
18.3^{x-3} + 39.3^{1+x} = 18 \cdot 3^{x-3} + 39 \cdot 3^{x+1} = 18 \cdot 3^{x-3} + 39 \cdot 3^{x} \cdot 3
\]
This can be factored as:
\[
3^{x-3}(18 + 39 \cdot 3^{4}) = 3^{x-3}(18 + 117) = 3^{x-3}(135)
\]
3. **Putting it all together:**
\[
\frac{3^{x}(-62)}{3^{x-3}(135)} = \frac{-62 \cdot 3^{3}}{135} = \frac{-62 \cdot 27}{135}
\]
4. **Final simplification:**
\[
\frac{-1674}{135} = \frac{-62}{5} \text{ (after reducing)}
\]
Thus, the simplified form is:
\[
\frac{100 \times 193^{x} - 93^{x+2}}{100000 \times 183^{x-3} + 10 \times 393^{x+1}}
\]
### 1.1.2 Simplifying \( \frac{\sqrt{48 x^{6}}-\sqrt[3]{125 x^{9}} \cdot 3^{\frac{1}{2}}}{\sqrt{75 x^{6}}} \)
1. **Calculate each term:**
- \( \sqrt{48 x^{6}} = \sqrt{48} \cdot x^{3} = 4\sqrt{3} \cdot x^{3} \)
- \( \sqrt[3]{125 x^{9}} = 5x^{3} \)
- Thus, \( \sqrt[3]{125 x^{9}} \cdot 3^{\frac{1}{2}} = 5x^{3} \cdot \sqrt{3} \)
2. **Combine the terms:**
\[
\frac{4\sqrt{3} x^{3} - 5x^{3} \sqrt{3}}{\sqrt{75 x^{6}}}
\]
The numerator simplifies to:
\[
(4 - 5)x^{3}\sqrt{3} = -x^{3}\sqrt{3}
\]
3. **Denominator:**
\[
\sqrt{75 x^{6}} = \sqrt{75} \cdot x^{3} = 5\sqrt{3} \cdot x^{3}
\]
4. **Final expression:**
\[
\frac{-x^{3}\sqrt{3}}{5\sqrt{3} \cdot x^{3}} = \frac{-1}{5}
\]
Thus, the simplified form is:
\[
-\frac{1}{5} \text{ or } -0.2
\]
### Summary of Results:
1.1.1: \( \frac{100 \times 193^{x} - 93^{x+2}}{100000 \times 183^{x-3} + 10 \times 393^{x+1}} \)
1.1.2: \( -\frac{1}{5} \) or \( -0.2 \)
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