Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine which expression is equivalent to [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex], let's go through the simplification step by step:
1. Start with the original expression:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} \][/tex]
2. Recall that the square root of a product can be expressed as the product of the square roots:
[tex]\[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \][/tex]
So we can write:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} = \sqrt{10^{\frac{3}{4}}} \cdot \sqrt{x} \][/tex]
3. Now, simplify [tex]\(\sqrt{10^{\frac{3}{4}}}\)[/tex]. The property of square roots and exponents tells us:
[tex]\[ \sqrt{a^b} = a^{b/2} \][/tex]
Therefore:
[tex]\[ \sqrt{10^{\frac{3}{4}}} = 10^{\frac{3}{4 \cdot 2}} = 10^{\frac{3}{8}} \][/tex]
4. Substitute back into our expression:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} = 10^{\frac{3}{8}} \cdot \sqrt{x} \][/tex]
5. Next, we look to match our simplified expression with one of the given choices. Let's transform each of the given choices and compare:
- [tex]\((\sqrt[3]{10})^{4 x}\)[/tex]:
[tex]\[ (\sqrt[3]{10})^{4 x} = (10^{\frac{1}{3}})^{4 x} = 10^{\frac{4 x}{3}} \][/tex]
- [tex]\((\sqrt[4]{10})^{3 x}\)[/tex]:
[tex]\[ (\sqrt[4]{10})^{3 x} = (10^{\frac{1}{4}})^{3 x} = 10^{\frac{3 x}{4}} \][/tex]
- [tex]\((\sqrt[6]{10})^{4 x}\)[/tex]:
[tex]\[ (\sqrt[6]{10})^{4 x} = (10^{\frac{1}{6}})^{4 x} = 10^{\frac{4 x}{6}} = 10^{\frac{2 x}{3}} \][/tex]
- [tex]\((\sqrt[8]{10})^{3 x}\)[/tex]:
[tex]\[ (\sqrt[8]{10})^{3 x} = (10^{\frac{1}{8}})^{3 x} = 10^{\frac{3 x}{8}} \][/tex]
6. Comparing these, we see that the expression [tex]\(10^{\frac{3 x}{8}}\)[/tex] matches our simplified form.
Therefore, the equivalent expression to [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex] is:
[tex]\[ (\sqrt[8]{10})^{3 x} \][/tex]
1. Start with the original expression:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} \][/tex]
2. Recall that the square root of a product can be expressed as the product of the square roots:
[tex]\[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \][/tex]
So we can write:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} = \sqrt{10^{\frac{3}{4}}} \cdot \sqrt{x} \][/tex]
3. Now, simplify [tex]\(\sqrt{10^{\frac{3}{4}}}\)[/tex]. The property of square roots and exponents tells us:
[tex]\[ \sqrt{a^b} = a^{b/2} \][/tex]
Therefore:
[tex]\[ \sqrt{10^{\frac{3}{4}}} = 10^{\frac{3}{4 \cdot 2}} = 10^{\frac{3}{8}} \][/tex]
4. Substitute back into our expression:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} = 10^{\frac{3}{8}} \cdot \sqrt{x} \][/tex]
5. Next, we look to match our simplified expression with one of the given choices. Let's transform each of the given choices and compare:
- [tex]\((\sqrt[3]{10})^{4 x}\)[/tex]:
[tex]\[ (\sqrt[3]{10})^{4 x} = (10^{\frac{1}{3}})^{4 x} = 10^{\frac{4 x}{3}} \][/tex]
- [tex]\((\sqrt[4]{10})^{3 x}\)[/tex]:
[tex]\[ (\sqrt[4]{10})^{3 x} = (10^{\frac{1}{4}})^{3 x} = 10^{\frac{3 x}{4}} \][/tex]
- [tex]\((\sqrt[6]{10})^{4 x}\)[/tex]:
[tex]\[ (\sqrt[6]{10})^{4 x} = (10^{\frac{1}{6}})^{4 x} = 10^{\frac{4 x}{6}} = 10^{\frac{2 x}{3}} \][/tex]
- [tex]\((\sqrt[8]{10})^{3 x}\)[/tex]:
[tex]\[ (\sqrt[8]{10})^{3 x} = (10^{\frac{1}{8}})^{3 x} = 10^{\frac{3 x}{8}} \][/tex]
6. Comparing these, we see that the expression [tex]\(10^{\frac{3 x}{8}}\)[/tex] matches our simplified form.
Therefore, the equivalent expression to [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex] is:
[tex]\[ (\sqrt[8]{10})^{3 x} \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.