Use The Properties Of Exponents To Determine The Value Of $a$ For The Equation:$\left(x^4\right)^{\frac{1}{2}} \sqrt[3]{x^4}=x^a$

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Introduction

Exponents are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will focus on using the properties of exponents to determine the value of $a$ for the equation: $\left(x4\right){\frac{1}{2}} \sqrt[3]{x4}=xa$. This equation involves the use of exponents, radicals, and properties of exponents, making it an excellent example of how to apply these concepts to solve a mathematical problem.

Understanding Exponents and Radicals

Before we dive into solving the equation, let's briefly review the concepts of exponents and radicals. Exponents are a shorthand way of representing repeated multiplication of a number. For example, $x^4$ means $x \times x \times x \times x$. Radicals, on the other hand, are a way of representing the nth root of a number. For example, $\sqrt[3]{x}$ means the cube root of $x$.

Properties of Exponents

Exponents have several properties that are essential for solving mathematical problems. Some of the key properties of exponents include:

  • Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, $x^a \times x^b = x^{a+b}$.
  • Power of a Power: When raising a power to another power, we multiply the exponents. For example, $(xa)b = x^{ab}$.
  • Quotient of Powers: When dividing two powers with the same base, we subtract the exponents. For example, $\frac{xa}{xb} = x^{a-b}$.

Solving the Equation

Now that we have reviewed the concepts of exponents and radicals, and the properties of exponents, let's focus on solving the equation: $\left(x4\right){\frac{1}{2}} \sqrt[3]{x4}=xa$. To solve this equation, we will apply the properties of exponents and radicals.

Step 1: Simplify the Left-Hand Side of the Equation

The left-hand side of the equation involves the product of two expressions: $\left(x4\right){\frac{1}{2}}$ and $\sqrt[3]{x^4}$. We can simplify this expression by applying the properties of exponents.

(x4)12=x4×12=x2\left(x^4\right)^{\frac{1}{2}} = x^{4 \times \frac{1}{2}} = x^2

x43=x43\sqrt[3]{x^4} = x^{\frac{4}{3}}

Therefore, the left-hand side of the equation becomes:

x2×x43=x2+43=x103x^2 \times x^{\frac{4}{3}} = x^{2 + \frac{4}{3}} = x^{\frac{10}{3}}

Step 2: Equate the Exponents

Now that we have simplified the left-hand side of the equation, we can equate the exponents to solve for $a$.

x103=xax^{\frac{10}{3}} = x^a

Since the bases are the same, we can equate the exponents:

103=a\frac{10}{3} = a

Therefore, the value of $a$ is $\frac{10}{3}$.

Conclusion

In this article, we have used the properties of exponents to determine the value of $a$ for the equation: $\left(x4\right){\frac{1}{2}} \sqrt[3]{x4}=xa$. We have applied the properties of exponents, including the product of powers, power of a power, and quotient of powers, to simplify the left-hand side of the equation and equate the exponents. The value of $a$ is $\frac{10}{3}$.

Final Thoughts

Exponents are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. By applying the properties of exponents, we can simplify complex expressions and solve equations involving exponents and radicals. In this article, we have demonstrated how to use the properties of exponents to solve a mathematical problem, and we hope that this example will be helpful for readers who are struggling with similar problems.

Additional Resources

For readers who want to learn more about exponents and radicals, we recommend the following resources:

  • Khan Academy: Exponents and Radicals
  • Mathway: Exponents and Radicals
  • Wolfram Alpha: Exponents and Radicals

These resources provide a comprehensive introduction to exponents and radicals, including their properties, rules, and applications. We hope that these resources will be helpful for readers who want to learn more about this topic.

Introduction

Exponents and radicals are fundamental concepts in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will address some of the most frequently asked questions about exponents and radicals, providing clear and concise answers to help readers better understand these concepts.

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

A: An exponent is a shorthand way of representing repeated multiplication of a number, while a radical is a way of representing the nth root of a number. For example, $x^4$ means $x \times x \times x \times x$, while $\sqrt[3]{x}$ means the cube root of $x$.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the following properties:

  • Product of Powers: When multiplying two powers with the same base, you add the exponents. For example, $x^a \times x^b = x^{a+b}$.
  • Power of a Power: When raising a power to another power, you multiply the exponents. For example, $(xa)b = x^{ab}$.
  • Quotient of Powers: When dividing two powers with the same base, you subtract the exponents. For example, $\frac{xa}{xb} = x^{a-b}$.

Q: How do I simplify an expression with radicals?

A: To simplify an expression with radicals, you can use the following properties:

  • Product of Radicals: When multiplying two radicals with the same index, you multiply the radicands. For example, $\sqrt[3]{x} \times \sqrt[3]{y} = \sqrt[3]{xy}$.
  • Power of a Radical: When raising a radical to another power, you multiply the index by the exponent. For example, $(\sqrt[3]{x})^2 = \sqrt[6]{x^2}$.
  • Quotient of Radicals: When dividing two radicals with the same index, you divide the radicands. For example, $\frac{\sqrt[3]{x}}{\sqrt[3]{y}} = \sqrt[3]{\frac{x}{y}}$.

Q: How do I solve an equation with exponents?

A: To solve an equation with exponents, you can use the following steps:

  1. Simplify the left-hand side of the equation by applying the properties of exponents.
  2. Equate the exponents to solve for the variable.
  3. Check your solution by plugging it back into the original equation.

Q: How do I solve an equation with radicals?

A: To solve an equation with radicals, you can use the following steps:

  1. Simplify the left-hand side of the equation by applying the properties of radicals.
  2. Equate the radicands to solve for the variable.
  3. Check your solution by plugging it back into the original equation.

Q: What are some common mistakes to avoid when working with exponents and radicals?

A: Some common mistakes to avoid when working with exponents and radicals include:

  • Forgetting to simplify the left-hand side of the equation: Make sure to simplify the left-hand side of the equation before equating the exponents or radicands.
  • Not checking your solution: Always check your solution by plugging it back into the original equation to make sure it is correct.
  • Not using the correct properties of exponents and radicals: Make sure to use the correct properties of exponents and radicals when simplifying expressions and solving equations.

Conclusion

Exponents and radicals are fundamental concepts in mathematics, and understanding their properties is crucial for solving various mathematical problems. By following the steps outlined in this article, you can simplify expressions and solve equations involving exponents and radicals. Remember to avoid common mistakes and always check your solution to ensure accuracy.

Additional Resources

For readers who want to learn more about exponents and radicals, we recommend the following resources:

  • Khan Academy: Exponents and Radicals
  • Mathway: Exponents and Radicals
  • Wolfram Alpha: Exponents and Radicals

These resources provide a comprehensive introduction to exponents and radicals, including their properties, rules, and applications. We hope that these resources will be helpful for readers who want to learn more about this topic.