Simplify The Expression:$\[ \left(125 X^6 Y^5\right)^{\frac{1}{3}} \\]

Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. In this case, we're given the expression $\left(125 x^6 y^5\right)^{\frac{1}{3}}$, and we're asked to simplify it. To start, let's break down the expression into its components. We have a base of $125 x^6 y^5$ and an exponent of $\frac{1}{3}$.

Applying the Power Rule of Exponents

The power rule of exponents states that for any numbers $a$ and $b$ and any integer $n$, $(a^b)^n = a^{b \cdot n}$. In our case, we can apply this rule to simplify the expression. We'll multiply the exponent $\frac{1}{3}$ by the exponents of the base, $6$ and $5$, to get the new exponents.

Simplifying the Expression

Using the power rule, we can rewrite the expression as:

$$\left(125 x^6 y^5\right)^{\frac{1}{3}} = 125^{\frac{1}{3}} \cdot \left(x^6\right)^{\frac{1}{3}} \cdot \left(y^5\right)^{\frac{1}{3}}$$

Evaluating the Exponents

Now, let's evaluate the exponents. We have $125^{\frac{1}{3}}$, which is equal to $5$ since $5^3 = 125$. We also have $\left(x^6\right)^{\frac{1}{3}}$, which is equal to $x^2$ since $\left(x^6\right)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2$. Finally, we have $\left(y^5\right)^{\frac{1}{3}}$, which is equal to $y^{\frac{5}{3}}$ since $\left(y^5\right)^{\frac{1}{3}} = y^{5 \cdot \frac{1}{3}} = y^{\frac{5}{3}}$.

Combining the Terms

Now that we've evaluated the exponents, we can combine the terms to get the simplified expression:

$$125^{\frac{1}{3}} \cdot \left(x^6\right)^{\frac{1}{3}} \cdot \left(y^5\right)^{\frac{1}{3}} = 5 \cdot x^2 \cdot y^{\frac{5}{3}}$$

Final Answer

The simplified expression is $5x^2y^{\frac{5}{3}}$.

Understanding the Concept

In this problem, we applied the power rule of exponents to simplify the given expression. We broke down the expression into its components, applied the power rule, and then evaluated the exponents to get the final answer. This problem demonstrates the importance of understanding the rules of exponentiation and how to apply them to simplify complex expressions.

Real-World Applications

The concept of simplifying expressions with exponents has many real-world applications. For example, in physics, we often encounter expressions with exponents that need to be simplified in order to solve problems. In engineering, we may need to simplify expressions with exponents in order to design and build complex systems. In finance, we may need to simplify expressions with exponents in order to calculate interest rates and investments.

Conclusion

In conclusion, simplifying expressions with exponents is an essential skill in mathematics and has many real-world applications. By understanding the rules of exponentiation and how to apply them, we can simplify complex expressions and solve problems in a variety of fields.

Frequently Asked Questions

• Q: What is the power rule of exponents? A: The power rule of exponents states that for any numbers $a$ and $b$ and any integer $n$, $(a^b)^n = a^{b \cdot n}$.
• Q: How do I apply the power rule of exponents? A: To apply the power rule, multiply the exponent by the exponents of the base.
• Q: What is the simplified expression for $\left(125 x^6 y^5\right)^{\frac{1}{3}}$? A: The simplified expression is $5x^2y^{\frac{5}{3}}$.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Final Thoughts

Simplifying expressions with exponents is an essential skill in mathematics and has many real-world applications. By understanding the rules of exponentiation and how to apply them, we can simplify complex expressions and solve problems in a variety of fields.

Q&A: Simplifying Expressions with Exponents

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any numbers $a$ and $b$ and any integer $n$, $(a^b)^n = a^{b \cdot n}$. This means that when we have an expression with an exponent raised to another power, we can simplify it by multiplying the exponents.

Q: How do I apply the power rule of exponents?

A: To apply the power rule, multiply the exponent by the exponents of the base. For example, if we have the expression $(x^2)^3$, we can simplify it by multiplying the exponents: $x^{2 \cdot 3} = x^6$.

Q: What is the simplified expression for $\left(125 x^6 y^5\right)^{\frac{1}{3}}$?

A: The simplified expression is $5x^2y^{\frac{5}{3}}$. To simplify this expression, we applied the power rule of exponents by multiplying the exponents of the base.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we can use the rule that $a^{-n} = \frac{1}{a^n}$. For example, if we have the expression $x^{-2}$, we can simplify it by rewriting it as $\frac{1}{x^2}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is in the denominator. For example, $x^2$ indicates that $x$ is raised to the power of 2, while $x^{-2}$ indicates that $x$ is in the denominator and is raised to the power of -2.

Q: How do I simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, we can use the rule that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if we have the expression $x^{\frac{1}{2}}$, we can simplify it by rewriting it as $\sqrt{x}$.

Q: What is the simplified expression for $\left(x^2 y^3\right)^{\frac{1}{4}}$?

A: The simplified expression is $xy^{\frac{3}{4}}$. To simplify this expression, we applied the power rule of exponents by multiplying the exponents of the base.

Q: How do I simplify expressions with multiple bases?

A: To simplify expressions with multiple bases, we can use the rule that $(ab)^n = a^n b^n$. For example, if we have the expression $(xy)^2$, we can simplify it by rewriting it as $x^2 y^2$.

Q: What is the simplified expression for $\left(2x^3 y^2\right)^{\frac{1}{2}}$?

A: The simplified expression is $2^{\frac{1}{2}} x^{\frac{3}{2}} y$. To simplify this expression, we applied the power rule of exponents by multiplying the exponents of the base.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Final Thoughts

Simplifying expressions with exponents is an essential skill in mathematics and has many real-world applications. By understanding the rules of exponentiation and how to apply them, we can simplify complex expressions and solve problems in a variety of fields.