Which Is Equivalent To 10 3 4 X \sqrt{10}^{\frac{3}{4} X} 10 ​ 4 3 ​ X ?A. ( 10 3 ) 4 X (\sqrt[3]{10})^{4 X} ( 3 10 ​ ) 4 X B. ( 10 4 ) 3 X (\sqrt[4]{10})^{3 X} ( 4 10 ​ ) 3 X C. ( 10 6 ) 4 X (\sqrt[6]{10})^{4 X} ( 6 10 ​ ) 4 X D. ( 10 8 ) 3 X (\sqrt[8]{10})^{3 X} ( 8 10 ​ ) 3 X

Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. One of the key concepts in this area is the ability to rewrite expressions in equivalent forms. In this article, we will explore how to simplify the expression $\sqrt{10}^{\frac{3}{4} x}$ and find its equivalent forms.

Understanding Exponents and Roots

Before we dive into the problem, let's quickly review the basics of exponents and roots. An exponent is a small number that is raised to a power, indicating how many times the base is multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$. A root, on the other hand, is the inverse operation of an exponent. For example, $\sqrt{8}$ means finding the number that, when multiplied by itself, equals $8$.

Simplifying the Expression

Now that we have a basic understanding of exponents and roots, let's simplify the expression $\sqrt{10}^{\frac{3}{4} x}$. To do this, we need to use the property of exponents that states $(a^m)^n = a^{m \cdot n}$. In this case, we can rewrite the expression as $(\sqrt{10})^{\frac{3}{4} x}$.

Next, we need to simplify the expression inside the parentheses. We can do this by using the property of roots that states $\sqrt[n]{a} = a^{\frac{1}{n}}$. In this case, we can rewrite the expression as $(10^{\frac{1}{2}})^{\frac{3}{4} x}$.

Now, we can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$ again to simplify the expression. This gives us $10^{\frac{1}{2} \cdot \frac{3}{4} x}$.

Evaluating the Exponent

Now that we have simplified the expression, let's evaluate the exponent. The exponent is $\frac{1}{2} \cdot \frac{3}{4} x$, which equals $\frac{3}{8} x$.

Finding the Equivalent Forms

Now that we have simplified the expression, let's find its equivalent forms. We can do this by using the property of exponents that states $a^{m \cdot n} = (a^m)^n$. In this case, we can rewrite the expression as $(10^{\frac{3}{8} x})^{\frac{2}{3}}$.

Comparing the Options

Now that we have found the equivalent forms of the expression, let's compare them to the options given in the problem. We can see that option A, $(\sqrt[3]{10})^{4 x}$, is not equivalent to the expression. However, option B, $(\sqrt[4]{10})^{3 x}$, is equivalent to the expression.

Conclusion

In conclusion, the expression $\sqrt{10}^{\frac{3}{4} x}$ is equivalent to $(\sqrt[4]{10})^{3 x}$. This is because we can rewrite the expression as $(10^{\frac{1}{2}})^{\frac{3}{4} x}$, which equals $10^{\frac{3}{8} x}$. We can then use the property of exponents that states $a^{m \cdot n} = (a^m)^n$ to rewrite the expression as $(10^{\frac{3}{8} x})^{\frac{2}{3}}$. This is equivalent to option B, $(\sqrt[4]{10})^{3 x}$.

Final Answer

Q: What is the difference between an exponent and a root?

A: An exponent is a small number that is raised to a power, indicating how many times the base is multiplied by itself. A root, on the other hand, is the inverse operation of an exponent. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$. A root, such as $\sqrt{8}$, means finding the number that, when multiplied by itself, equals $8$.

Q: How do I simplify an expression with a square root and an exponent?

A: To simplify an expression with a square root and an exponent, you can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$. You can also use the property of roots that states $\sqrt[n]{a} = a^{\frac{1}{n}}$. For example, to simplify the expression $\sqrt{10}^{\frac{3}{4} x}$, you can rewrite it as $(\sqrt{10})^{\frac{3}{4} x}$ and then simplify further using the properties of exponents and roots.

Q: What is the rule for multiplying exponents with the same base?

A: The rule for multiplying exponents with the same base is to add the exponents. For example, $a^m \cdot a^n = a^{m + n}$. This rule can be used to simplify expressions with multiple exponents.

Q: How do I simplify an expression with a fraction as an exponent?

A: To simplify an expression with a fraction as an exponent, you can use the property of exponents that states $a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}}$. For example, to simplify the expression $10^{\frac{3}{4} x}$, you can rewrite it as $(10^3)^{\frac{1}{4} x}$ and then simplify further using the properties of exponents.

Q: What is the difference between an exponential expression and a radical expression?

A: An exponential expression is an expression that involves raising a number to a power, such as $2^3$. A radical expression, on the other hand, is an expression that involves finding the root of a number, such as $\sqrt{8}$. While both types of expressions involve powers and roots, they are distinct and have different properties.

Q: How do I simplify an expression with multiple exponents and roots?

A: To simplify an expression with multiple exponents and roots, you can use the properties of exponents and roots to rewrite the expression in a simpler form. For example, to simplify the expression $\sqrt{10}^{\frac{3}{4} x}$, you can rewrite it as $(\sqrt{10})^{\frac{3}{4} x}$ and then simplify further using the properties of exponents and roots.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is option B, $(\sqrt[4]{10})^{3 x}$. This is because we can rewrite the expression $\sqrt{10}^{\frac{3}{4} x}$ as $(\sqrt{10})^{\frac{3}{4} x}$, which equals $10^{\frac{3}{8} x}$. We can then use the property of exponents that states $a^{m \cdot n} = (a^m)^n$ to rewrite the expression as $(10^{\frac{3}{8} x})^{\frac{2}{3}}$, which is equivalent to option B.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. By using the properties of exponents and roots, we can rewrite expressions in simpler forms and find their equivalent forms. We hope that this article has provided you with a better understanding of how to simplify exponential expressions and has helped you to develop your problem-solving skills.