Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To simplify the given expression [tex]\(\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}\)[/tex], we will break it down step by step.
1. Simplify the Cube Roots Separately:
First, let's simplify the numerators and denominators inside the cube roots.
The numerator is [tex]\(\sqrt[3]{32 x^3 y^6}\)[/tex]:
[tex]\[ \sqrt[3]{32 x^3 y^6} = 32^{1/3} \cdot (x^3)^{1/3} \cdot (y^6)^{1/3} \][/tex]
Simplifying each term individually:
[tex]\[ 32^{1/3} = 2^{5/3} = 2 \cdot 2^{2/3} \][/tex]
[tex]\[ (x^3)^{1/3} = x \][/tex]
[tex]\[ (y^6)^{1/3} = y^2 \][/tex]
So,
[tex]\[ \sqrt[3]{32 x^3 y^6} = 2 \cdot x \cdot y^2 \cdot 2^{2/3} = 2 \cdot 2^{2/3} \cdot x \cdot y^2 \][/tex]
Next, let's simplify the denominator [tex]\(\sqrt[3]{2 x^9 y^2}\)[/tex]:
[tex]\[ \sqrt[3]{2 x^9 y^2} = (2)^{1/3} \cdot (x^9)^{1/3} \cdot (y^2)^{1/3} \][/tex]
Simplifying each term individually:
[tex]\[ 2^{1/3} \][/tex]
[tex]\[ (x^9)^{1/3} = x^3 \][/tex]
[tex]\[ (y^2)^{1/3} = y^{2/3} \][/tex]
So,
[tex]\[ \sqrt[3]{2 x^9 y^2} = 2^{1/3} \cdot x^3 \cdot y^{2/3} \][/tex]
2. Combine the Simplified Forms:
Now we put the simplified numerator and denominator together:
[tex]\[ \frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = \frac{2 \cdot 2^{2/3} \cdot x \cdot y^2}{2^{1/3} \cdot x^3 \cdot y^{2/3}} \][/tex]
3. Simplify the Fraction:
We can simplify this by canceling like terms and combining the powers of 2:
[tex]\[ \frac{2 \cdot 2^{2/3}}{2^{1/3}} = 2^{1 + 2/3 - 1/3} = 2^{4/3} \][/tex]
For the [tex]\(x\)[/tex] terms:
[tex]\[ \frac{x}{x^3} = x^{1-3} = x^{-2} \][/tex]
For the [tex]\(y\)[/tex] terms:
[tex]\[ \frac{y^2}{y^{2/3}} = y^{2 - 2/3} = y^{4/3} \][/tex]
Thus, the simplified form is:
[tex]\[ 2^{4/3} \cdot x^{-2} \cdot y^{4/3} \][/tex]
But [tex]\(2^{4/3} = (2^{1/3})^4\)[/tex], simplifying it separately:
[tex]\[ (2^{1/3})^4 = 2^{4/3} \][/tex]
Hence, the expression becomes:
[tex]\[ \frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = 2^{4/3} \cdot x^{-2} \cdot y^{4/3} \][/tex]
Therefore, the equivalent simplified form of [tex]\(\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}\)[/tex] is:
[tex]\[ 2^{4/3} \cdot \frac{y^{4/3}}{x^2} \][/tex]
1. Simplify the Cube Roots Separately:
First, let's simplify the numerators and denominators inside the cube roots.
The numerator is [tex]\(\sqrt[3]{32 x^3 y^6}\)[/tex]:
[tex]\[ \sqrt[3]{32 x^3 y^6} = 32^{1/3} \cdot (x^3)^{1/3} \cdot (y^6)^{1/3} \][/tex]
Simplifying each term individually:
[tex]\[ 32^{1/3} = 2^{5/3} = 2 \cdot 2^{2/3} \][/tex]
[tex]\[ (x^3)^{1/3} = x \][/tex]
[tex]\[ (y^6)^{1/3} = y^2 \][/tex]
So,
[tex]\[ \sqrt[3]{32 x^3 y^6} = 2 \cdot x \cdot y^2 \cdot 2^{2/3} = 2 \cdot 2^{2/3} \cdot x \cdot y^2 \][/tex]
Next, let's simplify the denominator [tex]\(\sqrt[3]{2 x^9 y^2}\)[/tex]:
[tex]\[ \sqrt[3]{2 x^9 y^2} = (2)^{1/3} \cdot (x^9)^{1/3} \cdot (y^2)^{1/3} \][/tex]
Simplifying each term individually:
[tex]\[ 2^{1/3} \][/tex]
[tex]\[ (x^9)^{1/3} = x^3 \][/tex]
[tex]\[ (y^2)^{1/3} = y^{2/3} \][/tex]
So,
[tex]\[ \sqrt[3]{2 x^9 y^2} = 2^{1/3} \cdot x^3 \cdot y^{2/3} \][/tex]
2. Combine the Simplified Forms:
Now we put the simplified numerator and denominator together:
[tex]\[ \frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = \frac{2 \cdot 2^{2/3} \cdot x \cdot y^2}{2^{1/3} \cdot x^3 \cdot y^{2/3}} \][/tex]
3. Simplify the Fraction:
We can simplify this by canceling like terms and combining the powers of 2:
[tex]\[ \frac{2 \cdot 2^{2/3}}{2^{1/3}} = 2^{1 + 2/3 - 1/3} = 2^{4/3} \][/tex]
For the [tex]\(x\)[/tex] terms:
[tex]\[ \frac{x}{x^3} = x^{1-3} = x^{-2} \][/tex]
For the [tex]\(y\)[/tex] terms:
[tex]\[ \frac{y^2}{y^{2/3}} = y^{2 - 2/3} = y^{4/3} \][/tex]
Thus, the simplified form is:
[tex]\[ 2^{4/3} \cdot x^{-2} \cdot y^{4/3} \][/tex]
But [tex]\(2^{4/3} = (2^{1/3})^4\)[/tex], simplifying it separately:
[tex]\[ (2^{1/3})^4 = 2^{4/3} \][/tex]
Hence, the expression becomes:
[tex]\[ \frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = 2^{4/3} \cdot x^{-2} \cdot y^{4/3} \][/tex]
Therefore, the equivalent simplified form of [tex]\(\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}\)[/tex] is:
[tex]\[ 2^{4/3} \cdot \frac{y^{4/3}}{x^2} \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
