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Sagot :
Certainly! Let's delve into the problem step-by-step and explore the given expressions one by one.
### Expression Analysis
1. Expression 1:
[tex]\[ \sqrt[3]{875 x^5 y^9} \][/tex]
2. Expression 2:
[tex]\[ 5 x y^3 \cdot\left(7 x^2\right)^{\frac{1}{3}} \][/tex]
3. Expression 3:
[tex]\[ (125)^{\frac{1}{3}} \cdot(7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3}+\frac{2}{3}\right)} \cdot y^3 \][/tex]
### Step-by-Step Solution
#### Simplifying Expression 1
[tex]\[ \sqrt[3]{875 x^5 y^9} \][/tex]
To simplify, let's break it down term-by-term inside the cube root:
[tex]\[ 875 = 5^3 \cdot 7 \][/tex]
Thus, we get:
[tex]\[ \sqrt[3]{(5^3 \cdot 7) \cdot x^5 \cdot y^9} \][/tex]
Which can be expressed as:
[tex]\[ = \sqrt[3]{5^3 \cdot 7 \cdot x^5 \cdot y^9} \][/tex]
Using exponent rules:
[tex]\[ = \sqrt[3]{5^3} \cdot \sqrt[3]{7} \cdot \sqrt[3]{x^5} \cdot \sqrt[3]{y^9} \][/tex]
Calculating each cube root individually:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^3 \][/tex]
Given [tex]\( x = 5 \)[/tex] and [tex]\( y = 9 \)[/tex]:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot (5)^{\frac{5}{3}} \cdot (9)^3 \][/tex]
Hence:
[tex]\[ = 101940.52984716721 \][/tex]
#### Simplifying Expression 2
[tex]\[ 5 x y^3 \cdot\left(7 x^2\right)^{\frac{1}{3}} \][/tex]
Breaking it down step-by-step:
[tex]\[ = 5 x y^3 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{2}{3}} \][/tex]
Combining similar terms:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot x^{1 + \frac{2}{3}} \cdot y^3 \][/tex]
Given [tex]\( x = 5 \)[/tex] and [tex]\( y = 9 \)[/tex]:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot (5)^{\frac{5}{3}} \cdot (9)^3 \][/tex]
Thus:
[tex]\[ = 18225 \][/tex]
#### Expression 3
[tex]\[ (125)^{\frac{1}{3}} \cdot(7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3}+\frac{2}{3}\right)} \cdot y^3 \][/tex]
Breaking it down:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^3 \][/tex]
Given [tex]\( x = 5 \)[/tex] and [tex]\( y = 9 \)[/tex]:
[tex]\[ = (125 \cdot 7)^\frac{1}{3} \cdot (5)^{\frac{5}{3}} \cdot (9)^3 \][/tex]
Which results in:
[tex]\[ = 101940.52984716729 \][/tex]
#### Additional Steps Derived from Intermediate Results:
[tex]\[ 125^{\frac{1}{3}} = 4.999999999999999 \][/tex]
[tex]\[ 7^{\frac{1}{3}} = 1.912931182772389 \][/tex]
[tex]\[ x^{1 + \frac{2}{3}} = 14.620088691064327 \][/tex]
### Conclusion
Thus, the correct and simplified results for each expression are:
1. [tex]\(\sqrt[3]{875 x^5 y^9} = 101940.52984716721\)[/tex]
2. [tex]\(5 x y^3 \left(7 x^2\right)^{\frac{1}{3}} = 18225\)[/tex]
3. [tex]\( (125)^{\frac{1}{3}}(7)^{\frac{1}{3}} x^{\left(\frac{3}{3}+\frac{2}{3}\right)} y^3 = 101940.52984716729 \)[/tex]
And intermediate results:
- [tex]\((125)^{\frac{1}{3}} = 4.999999999999999\)[/tex]
- [tex]\((7)^{\frac{1}{3}} = 1.912931182772389\)[/tex]
- [tex]\( x^{\left(\frac{3}{3}+\frac{2}{3}\right)} = 14.620088691064327\)[/tex]
### Expression Analysis
1. Expression 1:
[tex]\[ \sqrt[3]{875 x^5 y^9} \][/tex]
2. Expression 2:
[tex]\[ 5 x y^3 \cdot\left(7 x^2\right)^{\frac{1}{3}} \][/tex]
3. Expression 3:
[tex]\[ (125)^{\frac{1}{3}} \cdot(7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3}+\frac{2}{3}\right)} \cdot y^3 \][/tex]
### Step-by-Step Solution
#### Simplifying Expression 1
[tex]\[ \sqrt[3]{875 x^5 y^9} \][/tex]
To simplify, let's break it down term-by-term inside the cube root:
[tex]\[ 875 = 5^3 \cdot 7 \][/tex]
Thus, we get:
[tex]\[ \sqrt[3]{(5^3 \cdot 7) \cdot x^5 \cdot y^9} \][/tex]
Which can be expressed as:
[tex]\[ = \sqrt[3]{5^3 \cdot 7 \cdot x^5 \cdot y^9} \][/tex]
Using exponent rules:
[tex]\[ = \sqrt[3]{5^3} \cdot \sqrt[3]{7} \cdot \sqrt[3]{x^5} \cdot \sqrt[3]{y^9} \][/tex]
Calculating each cube root individually:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^3 \][/tex]
Given [tex]\( x = 5 \)[/tex] and [tex]\( y = 9 \)[/tex]:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot (5)^{\frac{5}{3}} \cdot (9)^3 \][/tex]
Hence:
[tex]\[ = 101940.52984716721 \][/tex]
#### Simplifying Expression 2
[tex]\[ 5 x y^3 \cdot\left(7 x^2\right)^{\frac{1}{3}} \][/tex]
Breaking it down step-by-step:
[tex]\[ = 5 x y^3 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{2}{3}} \][/tex]
Combining similar terms:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot x^{1 + \frac{2}{3}} \cdot y^3 \][/tex]
Given [tex]\( x = 5 \)[/tex] and [tex]\( y = 9 \)[/tex]:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot (5)^{\frac{5}{3}} \cdot (9)^3 \][/tex]
Thus:
[tex]\[ = 18225 \][/tex]
#### Expression 3
[tex]\[ (125)^{\frac{1}{3}} \cdot(7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3}+\frac{2}{3}\right)} \cdot y^3 \][/tex]
Breaking it down:
[tex]\[ = 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^3 \][/tex]
Given [tex]\( x = 5 \)[/tex] and [tex]\( y = 9 \)[/tex]:
[tex]\[ = (125 \cdot 7)^\frac{1}{3} \cdot (5)^{\frac{5}{3}} \cdot (9)^3 \][/tex]
Which results in:
[tex]\[ = 101940.52984716729 \][/tex]
#### Additional Steps Derived from Intermediate Results:
[tex]\[ 125^{\frac{1}{3}} = 4.999999999999999 \][/tex]
[tex]\[ 7^{\frac{1}{3}} = 1.912931182772389 \][/tex]
[tex]\[ x^{1 + \frac{2}{3}} = 14.620088691064327 \][/tex]
### Conclusion
Thus, the correct and simplified results for each expression are:
1. [tex]\(\sqrt[3]{875 x^5 y^9} = 101940.52984716721\)[/tex]
2. [tex]\(5 x y^3 \left(7 x^2\right)^{\frac{1}{3}} = 18225\)[/tex]
3. [tex]\( (125)^{\frac{1}{3}}(7)^{\frac{1}{3}} x^{\left(\frac{3}{3}+\frac{2}{3}\right)} y^3 = 101940.52984716729 \)[/tex]
And intermediate results:
- [tex]\((125)^{\frac{1}{3}} = 4.999999999999999\)[/tex]
- [tex]\((7)^{\frac{1}{3}} = 1.912931182772389\)[/tex]
- [tex]\( x^{\left(\frac{3}{3}+\frac{2}{3}\right)} = 14.620088691064327\)[/tex]
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