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

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Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. The expression $\left(-2 y^5 x^4 \cdot x^5\right)^5$ involves both multiplication and exponentiation. To simplify this expression, we need to apply the rules of exponentiation, specifically the power of a product rule.

The Power of a Product Rule

The power of a product rule states that when a product of factors is raised to a power, each factor in the product is raised to that power. Mathematically, this can be represented as:

$(ab)^n = a^n \cdot b^n$

In the given expression, we have $\left(-2 y^5 x^4 \cdot x^5\right)^5$. Using the power of a product rule, we can rewrite this expression as:

$\left(-2 y^5 x^4\right)^5 \cdot \left(x^5\right)^5$

Applying the Power of a Product Rule

Now that we have applied the power of a product rule, we can simplify each factor raised to the power of 5.

Simplifying the First Factor

The first factor is $\left(-2 y^5 x^4\right)^5$. To simplify this, we need to apply the power of a power rule, which states that when a power is raised to another power, the exponents are multiplied. Mathematically, this can be represented as:

$\left(a^m\right)^n = a^{m \cdot n}$

Using this rule, we can rewrite the first factor as:

$(-2)^5 \cdot \left(y^5\right)^5 \cdot \left(x^4\right)^5$

Simplifying the Second Factor

The second factor is $\left(x^5\right)^5$. Using the power of a power rule, we can rewrite this factor as:

$x^{5 \cdot 5}$

Simplifying the Expression

Now that we have simplified each factor, we can rewrite the original expression as:

$(-2)^5 \cdot \left(y^5\right)^5 \cdot \left(x^4\right)^5 \cdot x^{5 \cdot 5}$

Evaluating the Expression

To evaluate the expression, we need to calculate the values of each factor.

Evaluating the First Factor

The first factor is $(-2)^5$. To evaluate this, we need to raise -2 to the power of 5.

$(-2)^5 = -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 = -32$

Evaluating the Second Factor

The second factor is $\left(y^5\right)^5$. To evaluate this, we need to raise $y^5$ to the power of 5.

$\left(y^5\right)^5 = y^{5 \cdot 5} = y^{25}$

Evaluating the Third Factor

The third factor is $\left(x^4\right)^5$. To evaluate this, we need to raise $x^4$ to the power of 5.

$\left(x^4\right)^5 = x^{4 \cdot 5} = x^{20}$

Evaluating the Fourth Factor

The fourth factor is $x^{5 \cdot 5}$. To evaluate this, we need to raise $x$ to the power of 25.

$x^{5 \cdot 5} = x^{25}$

Combining the Factors

Now that we have evaluated each factor, we can combine them to get the final result.

$(-2)^5 \cdot \left(y^5\right)^5 \cdot \left(x^4\right)^5 \cdot x^{5 \cdot 5} = -32 \cdot y^{25} \cdot x^{20} \cdot x^{25}$

Simplifying the Final Result

To simplify the final result, we can combine the like terms.

$-32 \cdot y^{25} \cdot x^{20} \cdot x^{25} = -32 \cdot y^{25} \cdot x^{45}$

The final answer is $\boxed{-32 y^{25} x^{45}}$.

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Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. The expression $\left(-2 y^5 x^4 \cdot x^5\right)^5$ involves both multiplication and exponentiation. To simplify this expression, we need to apply the rules of exponentiation, specifically the power of a product rule.

Q&A

Q: What is the power of a product rule?

A: The power of a product rule states that when a product of factors is raised to a power, each factor in the product is raised to that power. Mathematically, this can be represented as:

$(ab)^n = a^n \cdot b^n$

Q: How do I apply the power of a product rule to the given expression?

A: To apply the power of a product rule, we need to rewrite the expression as a product of factors. In this case, we can rewrite the expression as:

$\left(-2 y^5 x^4\right)^5 \cdot \left(x^5\right)^5$

Q: What is the power of a power rule?

A: The power of a power rule states that when a power is raised to another power, the exponents are multiplied. Mathematically, this can be represented as:

$\left(a^m\right)^n = a^{m \cdot n}$

Q: How do I apply the power of a power rule to the given expression?

A: To apply the power of a power rule, we need to rewrite each factor raised to the power of 5. In this case, we can rewrite the first factor as:

$(-2)^5 \cdot \left(y^5\right)^5 \cdot \left(x^4\right)^5$

Q: What is the final result of simplifying the expression?

A: The final result of simplifying the expression is:

$-32 \cdot y^{25} \cdot x^{45}$

Q: How do I evaluate the expression?

A: To evaluate the expression, we need to calculate the values of each factor. In this case, we need to raise -2 to the power of 5, $y^5$ to the power of 5, $x^4$ to the power of 5, and $x$ to the power of 25.

Q: What are the values of each factor?

A: The values of each factor are:

• $(-2)^5 = -32$
• $\left(y^5\right)^5 = y^{25}$
• $\left(x^4\right)^5 = x^{20}$
• $x^{5 \cdot 5} = x^{25}$

Q: How do I combine the factors to get the final result?

A: To combine the factors, we need to multiply them together. In this case, we can combine the factors as follows:

$-32 \cdot y^{25} \cdot x^{20} \cdot x^{25} = -32 \cdot y^{25} \cdot x^{45}$

Conclusion

Simplifying the expression $\left(-2 y^5 x^4 \cdot x^5\right)^5$ involves applying the rules of exponentiation, specifically the power of a product rule and the power of a power rule. By following these rules and evaluating each factor, we can simplify the expression to get the final result.

Common Mistakes

• Failing to apply the power of a product rule
• Failing to apply the power of a power rule
• Not evaluating each factor correctly
• Not combining the factors correctly

Tips and Tricks

• Make sure to apply the power of a product rule and the power of a power rule correctly
• Evaluate each factor carefully and correctly
• Combine the factors correctly to get the final result

Practice Problems

• Simplify the expression $\left(3 x^2 y^3 \cdot x^4\right)^3$
• Simplify the expression $\left(2 x^5 y^2 \cdot y^3\right)^2$
• Simplify the expression $\left(4 x^3 y^4 \cdot x^2\right)^4$

Resources

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