Express The Given Mathematical Expression In Simpler Terms Or Provide Its Context.d) $4xy^{3/4}$

Understanding the Mathematical Expression

The given mathematical expression is $4xy^{3/4}$. This expression involves a variable $x$, a variable $y$, and an exponent of $\frac{3}{4}$. To simplify this expression, we need to understand the properties of exponents and how to manipulate them.

Properties of Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ means $x \times x \times x$. When we have an exponent of $\frac{3}{4}$, it means that the base (in this case, $y$) is being raised to the power of $\frac{3}{4}$.

Simplifying the Expression

To simplify the expression $4xy^{3/4}$, we can start by rewriting the exponent as a fraction. We can do this by using the property of exponents that states $a^{m/n} = \sqrt[n]{a^m}$. In this case, we can rewrite $y^{3/4}$ as $\sqrt[4]{y^3}$.

4xy^{3/4} = 4x\sqrt[4]{y^3}

Rationalizing the Denominator

The expression $\sqrt[4]{y^3}$ can be further simplified by rationalizing the denominator. To do this, we need to multiply the numerator and denominator by a value that will eliminate the radical in the denominator.

\sqrt[4]{y^3} = \frac{\sqrt[4]{y^3} \times \sqrt[4]{y^3}}{\sqrt[4]{y^3} \times \sqrt[4]{y^3}} = \frac{y^{3/4} \times y^{3/4}}{y^{3/4} \times y^{3/4}} = \frac{y^{3/2}}{y^{3/4} \times y^{3/4}}

Simplifying the Expression Further

Now that we have rationalized the denominator, we can simplify the expression further by combining the terms.

4x\sqrt[4]{y^3} = 4x\frac{y^{3/2}}{y^{3/4} \times y^{3/4}} = 4x\frac{y^{3/2}}{y^{3/2}} = 4x

Conclusion

In conclusion, the mathematical expression $4xy^{3/4}$ can be simplified to $4x$. This is because the exponent of $\frac{3}{4}$ can be rewritten as a fraction, and the expression can be further simplified by rationalizing the denominator and combining the terms.

Context of the Mathematical Expression

The mathematical expression $4xy^{3/4}$ can be used in a variety of contexts, such as:

• Algebra: The expression can be used to simplify algebraic expressions and equations.
• Calculus: The expression can be used to find the derivative of a function.
• Physics: The expression can be used to describe the motion of an object.

Real-World Applications

The mathematical expression $4xy^{3/4}$ has real-world applications in various fields, such as:

• Engineering: The expression can be used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: The expression can be used to develop algorithms and data structures.
• Economics: The expression can be used to model economic systems and make predictions about economic trends.

Final Thoughts

Frequently Asked Questions

Q: What is the simplified form of the mathematical expression $4xy^{3/4}$?

A: The simplified form of the mathematical expression $4xy^{3/4}$ is $4x$.

Q: How do you simplify the expression $4xy^{3/4}$?

A: To simplify the expression $4xy^{3/4}$, you can start by rewriting the exponent as a fraction. You can do this by using the property of exponents that states $a^{m/n} = \sqrt[n]{a^m}$. In this case, you can rewrite $y^{3/4}$ as $\sqrt[4]{y^3}$.

Q: What is the property of exponents that allows you to rewrite $y^{3/4}$ as $\sqrt[4]{y^3}$?

A: The property of exponents that allows you to rewrite $y^{3/4}$ as $\sqrt[4]{y^3}$ is $a^{m/n} = \sqrt[n]{a^m}$.

Q: How do you rationalize the denominator of the expression $\sqrt[4]{y^3}$?

A: To rationalize the denominator of the expression $\sqrt[4]{y^3}$, you need to multiply the numerator and denominator by a value that will eliminate the radical in the denominator. In this case, you can multiply the numerator and denominator by $\sqrt[4]{y^3}$.

Q: What is the result of rationalizing the denominator of the expression $\sqrt[4]{y^3}$?

A: The result of rationalizing the denominator of the expression $\sqrt[4]{y^3}$ is $\frac{y^{3/2}}{y^{3/4} \times y^{3/4}}$.

Q: How do you simplify the expression $\frac{y^{3/2}}{y^{3/4} \times y^{3/4}}$?

A: To simplify the expression $\frac{y^{3/2}}{y^{3/4} \times y^{3/4}}$, you can combine the terms in the numerator and denominator.

Q: What is the final simplified form of the expression $4xy^{3/4}$?

A: The final simplified form of the expression $4xy^{3/4}$ is $4x$.

Q: What are some real-world applications of the mathematical expression $4xy^{3/4}$?

A: Some real-world applications of the mathematical expression $4xy^{3/4}$ include:

• Algebra: The expression can be used to simplify algebraic expressions and equations.
• Calculus: The expression can be used to find the derivative of a function.
• Physics: The expression can be used to describe the motion of an object.
• Engineering: The expression can be used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: The expression can be used to develop algorithms and data structures.
• Economics: The expression can be used to model economic systems and make predictions about economic trends.

Q: What are some tips for simplifying complex mathematical expressions?

A: Some tips for simplifying complex mathematical expressions include:

• Start by rewriting the expression in a simpler form: This can involve using properties of exponents, such as $a^{m/n} = \sqrt[n]{a^m}$.
• Use algebraic manipulations: This can involve combining like terms, factoring, and canceling out common factors.
• Look for patterns and relationships: This can involve recognizing that certain expressions are equivalent or can be simplified in a particular way.
• Use technology: This can involve using calculators or computer software to simplify complex expressions.

Q: What are some common mistakes to avoid when simplifying mathematical expressions?

A: Some common mistakes to avoid when simplifying mathematical expressions include:

• Not following the order of operations: This can involve not evaluating expressions in the correct order, such as not evaluating exponents before multiplication.
• Not using the correct properties of exponents: This can involve not using properties such as $a^{m/n} = \sqrt[n]{a^m}$ correctly.
• Not simplifying expressions fully: This can involve not simplifying expressions as much as possible, such as not combining like terms.
• Not checking for errors: This can involve not double-checking expressions for errors, such as not checking for division by zero.