Engage with knowledgeable experts and get accurate answers on IDNLearn.com. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
Sagot :
To determine which expression is equivalent to [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex], we need to simplify this expression and compare it to the given options:
Let's start by simplifying [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex].
1. Simplify the expression inside the square root:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} \][/tex]
2. Use the property of exponents and square roots:
[tex]\[ \sqrt{a} = a^{\frac{1}{2}} \][/tex]
Apply this property:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} = (10^{\frac{3}{4}} x)^{\frac{1}{2}} \][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[ (10^{\frac{3}{4}} x)^{\frac{1}{2}} = 10^{\frac{3}{4} \cdot \frac{1}{2}} \cdot x^{\frac{1}{2}} \][/tex]
Simplify the exponent:
[tex]\[ 10^{\frac{3}{4} \cdot \frac{1}{2}} = 10^{\frac{3}{8}} \][/tex]
So the expression simplifies to:
[tex]\[ 10^{\frac{3}{8}} \cdot x^{\frac{1}{2}} \][/tex]
Now, let's compare each of the given expressions to find the match:
1. Option (a): [tex]\((\sqrt[3]{10})^{4 x}\)[/tex]
Simplify [tex]\(\sqrt[3]{10}\)[/tex]:
[tex]\[ \sqrt[3]{10} = 10^{\frac{1}{3}} \][/tex]
Raise this to the power [tex]\(4x\)[/tex]:
[tex]\[ (\sqrt[3]{10})^{4 x} = \left(10^{\frac{1}{3}}\right)^{4 x} = 10^{\frac{4 x}{3}} \][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
2. Option (b): [tex]\((\sqrt[4]{10})^{3 x}\)[/tex]
Simplify [tex]\(\sqrt[4]{10}\)[/tex]:
[tex]\[ \sqrt[4]{10} = 10^{\frac{1}{4}} \][/tex]
Raise this to the power [tex]\(3 x\)[/tex]:
[tex]\[ (\sqrt[4]{10})^{3 x} = \left(10^{\frac{1}{4}}\right)^{3 x} = 10^{\frac{3 x}{4}} \][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
3. Option (c): [tex]\((\sqrt[6]{10})^{4 x}\)[/tex]
Simplify [tex]\(\sqrt[6]{10}\)[/tex]:
[tex]\[ \sqrt[6]{10} = 10^{\frac{1}{6}} \][/tex]
Raise this to the power [tex]\(4 x\)[/tex]:
[tex]\[ (\sqrt[6]{10})^{4 x} = \left(10^{\frac{1}{6}}\right)^{4 x} = 10^{\frac{4 x}{6}} = 10^{\frac{2 x}{3}} \][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
4. Option (d): [tex]\((\sqrt[8]{10})^{3 x}\)[/tex]
Simplify [tex]\(\sqrt[8]{10}\)[/tex]:
[tex]\[ \sqrt[8]{10} = 10^{\frac{1}{8}} \][/tex]
Raise this to the power [tex]\(3 x\)[/tex]:
[tex]\[ (\sqrt[8]{10})^{3 x} = \left(10^{\frac{1}{8}}\right)^{3 x} = 10^{\frac{3 x}{8}} \][/tex]
This matches [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
Therefore, the expression [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex] is equivalent to [tex]\((\sqrt[8]{10})^{3 x}\)[/tex].
Thus, the correct answer is:
[tex]\[ (\sqrt[8]{10})^{3 x} \][/tex]
Let's start by simplifying [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex].
1. Simplify the expression inside the square root:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} \][/tex]
2. Use the property of exponents and square roots:
[tex]\[ \sqrt{a} = a^{\frac{1}{2}} \][/tex]
Apply this property:
[tex]\[ \sqrt{10^{\frac{3}{4}} x} = (10^{\frac{3}{4}} x)^{\frac{1}{2}} \][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[ (10^{\frac{3}{4}} x)^{\frac{1}{2}} = 10^{\frac{3}{4} \cdot \frac{1}{2}} \cdot x^{\frac{1}{2}} \][/tex]
Simplify the exponent:
[tex]\[ 10^{\frac{3}{4} \cdot \frac{1}{2}} = 10^{\frac{3}{8}} \][/tex]
So the expression simplifies to:
[tex]\[ 10^{\frac{3}{8}} \cdot x^{\frac{1}{2}} \][/tex]
Now, let's compare each of the given expressions to find the match:
1. Option (a): [tex]\((\sqrt[3]{10})^{4 x}\)[/tex]
Simplify [tex]\(\sqrt[3]{10}\)[/tex]:
[tex]\[ \sqrt[3]{10} = 10^{\frac{1}{3}} \][/tex]
Raise this to the power [tex]\(4x\)[/tex]:
[tex]\[ (\sqrt[3]{10})^{4 x} = \left(10^{\frac{1}{3}}\right)^{4 x} = 10^{\frac{4 x}{3}} \][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
2. Option (b): [tex]\((\sqrt[4]{10})^{3 x}\)[/tex]
Simplify [tex]\(\sqrt[4]{10}\)[/tex]:
[tex]\[ \sqrt[4]{10} = 10^{\frac{1}{4}} \][/tex]
Raise this to the power [tex]\(3 x\)[/tex]:
[tex]\[ (\sqrt[4]{10})^{3 x} = \left(10^{\frac{1}{4}}\right)^{3 x} = 10^{\frac{3 x}{4}} \][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
3. Option (c): [tex]\((\sqrt[6]{10})^{4 x}\)[/tex]
Simplify [tex]\(\sqrt[6]{10}\)[/tex]:
[tex]\[ \sqrt[6]{10} = 10^{\frac{1}{6}} \][/tex]
Raise this to the power [tex]\(4 x\)[/tex]:
[tex]\[ (\sqrt[6]{10})^{4 x} = \left(10^{\frac{1}{6}}\right)^{4 x} = 10^{\frac{4 x}{6}} = 10^{\frac{2 x}{3}} \][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
4. Option (d): [tex]\((\sqrt[8]{10})^{3 x}\)[/tex]
Simplify [tex]\(\sqrt[8]{10}\)[/tex]:
[tex]\[ \sqrt[8]{10} = 10^{\frac{1}{8}} \][/tex]
Raise this to the power [tex]\(3 x\)[/tex]:
[tex]\[ (\sqrt[8]{10})^{3 x} = \left(10^{\frac{1}{8}}\right)^{3 x} = 10^{\frac{3 x}{8}} \][/tex]
This matches [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
Therefore, the expression [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex] is equivalent to [tex]\((\sqrt[8]{10})^{3 x}\)[/tex].
Thus, the correct answer is:
[tex]\[ (\sqrt[8]{10})^{3 x} \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
