Simplify The Expression: ( X 2 ⋅ Y 5 ) 4 \left(x^2 \cdot Y^5\right)^4 ( X 2 ⋅ Y 5 ) 4

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Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression $\left(x^2 \cdot y^5\right)^4$ and we're asked to simplify it. To do this, we need to apply the rules of exponents, specifically the power of a product rule.

The Power of a Product Rule

The power of a product rule states that when we have a product of two or more variables raised to a power, we can distribute the exponent to each variable. In other words, if we have $\left(ab\right)^n$, then we can rewrite it as $a^n \cdot b^n$. This rule is essential in simplifying expressions involving exponents.

Applying the Power of a Product Rule

Now, let's apply the power of a product rule to the given expression. We have $\left(x^2 \cdot y^5\right)^4$, and we can rewrite it as $\left(x^2\right)^4 \cdot \left(y^5\right)^4$. This is because we're distributing the exponent 4 to each variable.

Simplifying the Expression

Now that we've applied the power of a product rule, we can simplify the expression further. We have $\left(x^2\right)^4 \cdot \left(y^5\right)^4$, and we can rewrite it as $x^{2 \cdot 4} \cdot y^{5 \cdot 4}$. This is because we're multiplying the exponents.

Simplifying the Exponents

Now, let's simplify the exponents. We have $x^{2 \cdot 4} \cdot y^{5 \cdot 4}$, and we can rewrite it as $x^8 \cdot y^{20}$. This is because we're multiplying the exponents.

The Final Answer

Therefore, the simplified expression is $x^8 \cdot y^{20}$. This is the final answer to the problem.

Understanding the Concept

In this problem, we applied the power of a product rule to simplify the expression. This rule is essential in dealing with exponents, and it's used extensively in algebra and other branches of mathematics. By understanding and applying this rule, we can simplify complex expressions and solve problems more efficiently.

Real-World Applications

The concept of exponents and the power of a product rule has numerous real-world applications. For example, in finance, exponents are used to calculate compound interest and investment returns. In science, exponents are used to describe the growth and decay of populations and chemical reactions. In engineering, exponents are used to design and optimize systems and structures.

Conclusion

In conclusion, simplifying the expression $\left(x^2 \cdot y^5\right)^4$ requires applying the power of a product rule. This rule states that when we have a product of two or more variables raised to a power, we can distribute the exponent to each variable. By understanding and applying this rule, we can simplify complex expressions and solve problems more efficiently. The concept of exponents and the power of a product rule has numerous real-world applications, and it's essential in dealing with algebra and other branches of mathematics.

Frequently Asked Questions

• Q: What is the power of a product rule? A: The power of a product rule states that when we have a product of two or more variables raised to a power, we can distribute the exponent to each variable.
• Q: How do we apply the power of a product rule? A: We apply the power of a product rule by distributing the exponent to each variable.
• Q: What are the real-world applications of exponents and the power of a product rule? A: Exponents and the power of a product rule have numerous real-world applications, including finance, science, and engineering.

Additional Resources

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

Final Thoughts

Simplifying the expression $\left(x^2 \cdot y^5\right)^4$ requires applying the power of a product rule. This rule is essential in dealing with exponents and algebra, and it has numerous real-world applications. By understanding and applying this rule, we can simplify complex expressions and solve problems more efficiently.

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Frequently Asked Questions

Q: What is the power of a product rule?

A: The power of a product rule states that when we have a product of two or more variables raised to a power, we can distribute the exponent to each variable. This rule is essential in simplifying expressions involving exponents.

Q: How do we apply the power of a product rule?

A: We apply the power of a product rule by distributing the exponent to each variable. For example, if we have $\left(ab\right)^n$, then we can rewrite it as $a^n \cdot b^n$.

Q: What are the real-world applications of exponents and the power of a product rule?

A: Exponents and the power of a product rule have numerous real-world applications, including finance, science, and engineering. For example, in finance, exponents are used to calculate compound interest and investment returns. In science, exponents are used to describe the growth and decay of populations and chemical reactions. In engineering, exponents are used to design and optimize systems and structures.

Q: How do we simplify expressions involving exponents?

A: To simplify expressions involving exponents, we need to apply the rules of exponents, including the power of a product rule. We also need to understand the concept of exponents and how they are used to describe the behavior of variables.

Q: What is the difference between a variable and a constant?

A: A variable is a value that can change, while a constant is a value that remains the same. In the expression $\left(x^2 \cdot y^5\right)^4$, $x$ and $y$ are variables, while the exponent 4 is a constant.

Q: How do we multiply exponents?

A: To multiply exponents, we add the exponents together. For example, if we have $x^a \cdot x^b$, then we can rewrite it as $x^{a+b}$.

Q: How do we divide exponents?

A: To divide exponents, we subtract the exponents. For example, if we have $x^a \div x^b$, then we can rewrite it as $x^{a-b}$.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, $x^0 = 1$.

Q: What is the negative exponent rule?

A: The negative exponent rule states that any non-zero number raised to a negative power is equal to the reciprocal of the number raised to the positive power. For example, $x^{-a} = \frac{1}{x^a}$.

Q: How do we simplify expressions involving negative exponents?

A: To simplify expressions involving negative exponents, we need to apply the negative exponent rule. We also need to understand the concept of negative exponents and how they are used to describe the behavior of variables.

Q: What is the product of powers rule?

A: The product of powers rule states that when we have a product of two or more variables raised to a power, we can multiply the exponents together. For example, if we have $x^a \cdot x^b$, then we can rewrite it as $x^{a+b}$.

Q: How do we apply the product of powers rule?

A: We apply the product of powers rule by multiplying the exponents together. For example, if we have $x^a \cdot x^b$, then we can rewrite it as $x^{a+b}$.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when we have a quotient of two or more variables raised to a power, we can subtract the exponents. For example, if we have $x^a \div x^b$, then we can rewrite it as $x^{a-b}$.

Q: How do we apply the quotient of powers rule?

A: We apply the quotient of powers rule by subtracting the exponents. For example, if we have $x^a \div x^b$, then we can rewrite it as $x^{a-b}$.

Additional Resources

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

Final Thoughts

Simplifying the expression $\left(x^2 \cdot y^5\right)^4$ requires applying the power of a product rule. This rule is essential in dealing with exponents and algebra, and it has numerous real-world applications. By understanding and applying this rule, we can simplify complex expressions and solve problems more efficiently.