Simplify The Expression: ( X − 2 Y 4 ) 2 ⋅ 2 X 2 \left(x^{-2} Y^4\right)^2 \cdot 2 X^2 ( X − 2 Y 4 ) 2 ⋅ 2 X 2

$$

Understanding the Problem

When dealing with algebraic expressions, simplification is a crucial step to make the expression more manageable and easier to work with. In this problem, we are given the expression $\left(x^{-2} y^4\right)^2 \cdot 2 x^2$ and we need to simplify it. Simplifying expressions involves combining like terms, removing any unnecessary parentheses, and rewriting the expression in a more compact form.

Applying the Power Rule

To simplify the given expression, we will start by applying the power rule of exponents. The power rule states that for any variables $a$ and $b$ and any integers $m$ and $n$, $\left(a^m b^n\right)^p = a^{mp} b^{np}$. In our case, we have $\left(x^{-2} y^4\right)^2$, which can be simplified using the power rule.

$\left(x^{-2} y^4\right)^2 = x^{-2 \cdot 2} y^{4 \cdot 2} = x^{-4} y^8$

Simplifying the Expression

Now that we have simplified the first part of the expression, we can rewrite the original expression as $x^{-4} y^8 \cdot 2 x^2$. To simplify this expression further, we can combine the like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have $x^{-4}$ and $x^2$, which are like terms.

$x^{-4} y^8 \cdot 2 x^2 = 2 x^{-4 + 2} y^8 = 2 x^{-2} y^8$

Removing Negative Exponents

In the simplified expression $2 x^{-2} y^8$, we have a negative exponent. Negative exponents can be removed by taking the reciprocal of the base and changing the sign of the exponent. In this case, we can remove the negative exponent by taking the reciprocal of $x$ and changing the sign of the exponent.

$2 x^{-2} y^8 = \frac{2 y^8}{x^2}$

Final Simplified Expression

After removing the negative exponent, we are left with the final simplified expression $\frac{2 y^8}{x^2}$. This is the simplest form of the original expression.

Conclusion

In this problem, we simplified the expression $\left(x^{-2} y^4\right)^2 \cdot 2 x^2$ using the power rule of exponents and combining like terms. We removed the negative exponent by taking the reciprocal of the base and changing the sign of the exponent. The final simplified expression is $\frac{2 y^8}{x^2}$.

Tips and Tricks

• When simplifying expressions, always start by applying the power rule of exponents.
• Combine like terms to simplify the expression further.
• Remove negative exponents by taking the reciprocal of the base and changing the sign of the exponent.

Common Mistakes to Avoid

• Failing to apply the power rule of exponents.
• Not combining like terms.
• Not removing negative exponents.

Real-World Applications

Simplifying expressions is an essential skill in mathematics and has many real-world applications. In physics, for example, simplifying expressions is used to solve problems involving motion and energy. In engineering, simplifying expressions is used to design and optimize systems.

Practice Problems

1. Simplify the expression $\left(x^3 y^2\right)^4 \cdot 3 x^2$.
2. Simplify the expression $\left(x^{-3} y^5\right)^3 \cdot 2 x^3$.
3. Simplify the expression $\left(x^2 y^4\right)^2 \cdot 4 x^3$.

Solutions

1. $\left(x^3 y^2\right)^4 \cdot 3 x^2 = 3 x^{8 + 2} y^{8} = 3 x^{10} y^8$
2. $\left(x^{-3} y^5\right)^3 \cdot 2 x^3 = 2 x^{-9 + 3} y^{15} = 2 x^{-6} y^{15} = \frac{2 y^{15}}{x^6}$
3. $\left(x^2 y^4\right)^2 \cdot 4 x^3 = 4 x^{4 + 3} y^{8} = 4 x^7 y^8$

Conclusion

Simplifying expressions is an essential skill in mathematics that has many real-world applications. By applying the power rule of exponents and combining like terms, we can simplify expressions and make them more manageable. In this problem, we simplified the expression $\left(x^{-2} y^4\right)^2 \cdot 2 x^2$ using the power rule of exponents and combining like terms. The final simplified expression is $\frac{2 y^8}{x^2}$.

$$

Understanding the Problem

When dealing with algebraic expressions, simplification is a crucial step to make the expression more manageable and easier to work with. In this problem, we are given the expression $\left(x^{-2} y^4\right)^2 \cdot 2 x^2$ and we need to simplify it. Simplifying expressions involves combining like terms, removing any unnecessary parentheses, and rewriting the expression in a more compact form.

Q&A

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any variables $a$ and $b$ and any integers $m$ and $n$, $\left(a^m b^n\right)^p = a^{mp} b^{np}$.

Q: How do I simplify the expression $\left(x^{-2} y^4\right)^2$?

A: To simplify the expression $\left(x^{-2} y^4\right)^2$, we can apply the power rule of exponents. This gives us $x^{-2 \cdot 2} y^{4 \cdot 2} = x^{-4} y^8$.

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

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

Q: How do I remove a negative exponent?

A: To remove a negative exponent, we can take the reciprocal of the base and change the sign of the exponent. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$.

Q: What is the final simplified expression for $\left(x^{-2} y^4\right)^2 \cdot 2 x^2$?

A: The final simplified expression for $\left(x^{-2} y^4\right)^2 \cdot 2 x^2$ is $\frac{2 y^8}{x^2}$.

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

A: Some common mistakes to avoid when simplifying expressions include failing to apply the power rule of exponents, not combining like terms, and not removing negative exponents.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including physics, engineering, and computer science. In physics, for example, simplifying expressions is used to solve problems involving motion and energy. In engineering, simplifying expressions is used to design and optimize systems.

Practice Problems

1. Simplify the expression $\left(x^3 y^2\right)^4 \cdot 3 x^2$.
2. Simplify the expression $\left(x^{-3} y^5\right)^3 \cdot 2 x^3$.
3. Simplify the expression $\left(x^2 y^4\right)^2 \cdot 4 x^3$.

Solutions

1. $\left(x^3 y^2\right)^4 \cdot 3 x^2 = 3 x^{8 + 2} y^{8} = 3 x^{10} y^8$
2. $\left(x^{-3} y^5\right)^3 \cdot 2 x^3 = 2 x^{-9 + 3} y^{15} = 2 x^{-6} y^{15} = \frac{2 y^{15}}{x^6}$
3. $\left(x^2 y^4\right)^2 \cdot 4 x^3 = 4 x^{4 + 3} y^{8} = 4 x^7 y^8$

Conclusion

Simplifying expressions is an essential skill in mathematics that has many real-world applications. By applying the power rule of exponents and combining like terms, we can simplify expressions and make them more manageable. In this problem, we simplified the expression $\left(x^{-2} y^4\right)^2 \cdot 2 x^2$ using the power rule of exponents and combining like terms. The final simplified expression is $\frac{2 y^8}{x^2}$.