Simplify Without A Negative Exponent: ( 2 X 2 Y ) 4 ( X 4 Y 5 ) 10 \left(2x^2 Y\right)^4 \left(x^4 Y^5\right)^{10} ( 2 X 2 Y ) 4 ( X 4 Y 5 ) 10

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Understanding the Problem

Simplifying Exponents is a crucial concept in mathematics, and it's essential to understand how to handle expressions with multiple exponents. In this article, we will focus on simplifying the given expression without a negative exponent: $\left(2x^2 y\right)^4 \left(x^4 y^5\right)^{10}$.

Applying the Power of a Product Rule

To simplify the given expression, we need to apply the Power of a Product Rule, which states that for any non-zero number $a$ and any integers $m$ and $n$, $\left(ab\right)^m = a^m b^m$. This rule allows us to distribute the exponent to each factor inside the parentheses.

Distributing Exponents

Using the Power of a Product Rule, we can rewrite the given expression as:

$\left(2x^2 y\right)^4 \left(x^4 y^5\right)^{10} = \left(2^4\right)\left(x^2\right)^4 \left(y\right)^4 \left(x^4\right)^{10} \left(y^5\right)^{10}$

Simplifying Exponents

Now, we can simplify each factor by applying the Power of a Power Rule, which states that for any non-zero number $a$ and any integers $m$ and $n$, $\left(a^m\right)^n = a^{mn}$. This rule allows us to multiply the exponents.

Applying the Power of a Power Rule

Using the Power of a Power Rule, we can rewrite each factor as:

$\left(2^4\right) = 2^4$ $\left(x^2\right)^4 = x^{2 \cdot 4} = x^8$ $\left(y\right)^4 = y^4$ $\left(x^4\right)^{10} = x^{4 \cdot 10} = x^{40}$ $\left(y^5\right)^{10} = y^{5 \cdot 10} = y^{50}$

Combining Like Terms

Now, we can combine like terms by adding the exponents of the same base.

Simplifying the Expression

Using the rule for combining like terms, we can rewrite the expression as:

$2^4 x^8 y^4 x^{40} y^{50} = 2^4 x^{8 + 40} y^{4 + 50}$

Final Simplification

Simplifying the expression further, we get:

$2^4 x^{48} y^{54} = 16 x^{48} y^{54}$

Conclusion

In this article, we have simplified the given expression without a negative exponent using the Power of a Product Rule and the Power of a Power Rule. We have also combined like terms to get the final simplified expression. This example demonstrates the importance of understanding and applying the rules of exponents in mathematics.

Frequently Asked Questions

• What is the Power of a Product Rule? The Power of a Product Rule states that for any non-zero number $a$ and any integers $m$ and $n$, $\left(ab\right)^m = a^m b^m$.
• What is the Power of a Power Rule? The Power of a Power Rule states that for any non-zero number $a$ and any integers $m$ and $n$, $\left(a^m\right)^n = a^{mn}$.
• How do I simplify expressions with multiple exponents? To simplify expressions with multiple exponents, you need to apply the Power of a Product Rule and the Power of a Power Rule, and then combine like terms.

Additional Resources

• Exponents and Powers: A comprehensive guide to exponents and powers, including rules and examples.
• Simplifying Expressions: A tutorial on simplifying expressions with multiple exponents.
• Mathematics Tutorials: A collection of tutorials on various mathematics topics, including exponents and powers.

Final Thoughts

Simplifying expressions with multiple exponents is an essential skill in mathematics. By understanding and applying the rules of exponents, you can simplify complex expressions and solve problems with ease. Remember to always apply the Power of a Product Rule and the Power of a Power Rule, and then combine like terms to get the final simplified expression.

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Q&A: Simplifying Exponents

In our previous article, we simplified the expression $\left(2x^2 y\right)^4 \left(x^4 y^5\right)^{10}$ using the Power of a Product Rule and the Power of a Power Rule. In this article, we will answer some frequently asked questions about simplifying exponents.

Q: What is the Power of a Product Rule?

A: The Power of a Product Rule states that for any non-zero number $a$ and any integers $m$ and $n$, $\left(ab\right)^m = a^m b^m$. This rule allows us to distribute the exponent to each factor inside the parentheses.

Q: What is the Power of a Power Rule?

A: The Power of a Power Rule states that for any non-zero number $a$ and any integers $m$ and $n$, $\left(a^m\right)^n = a^{mn}$. This rule allows us to multiply the exponents.

Q: How do I simplify expressions with multiple exponents?

A: To simplify expressions with multiple exponents, you need to apply the Power of a Product Rule and the Power of a Power Rule, and then combine like terms.

Q: What is the rule for combining like terms?

A: The rule for combining like terms states that for any non-zero number $a$ and any integers $m$ and $n$, $a^m a^n = a^{m+n}$. This rule allows us to add the exponents of the same base.

Q: Can I simplify expressions with negative exponents?

A: Yes, you can simplify expressions with negative exponents by applying the rule for negative exponents, which states that for any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$.

Q: How do I handle expressions with zero exponents?

A: Expressions with zero exponents are equal to 1, so you can simplify them by setting the exponent to 0.

Q: Can I simplify expressions with fractional exponents?

A: Yes, you can simplify expressions with fractional exponents by applying the rule for fractional exponents, which states that for any non-zero number $a$ and any integers $m$ and $n$, $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: How do I simplify expressions with multiple bases?

A: To simplify expressions with multiple bases, you need to apply the rule for multiplying powers with the same base, which states that for any non-zero number $a$ and any integers $m$ and $n$, $a^m a^n = a^{m+n}$.

Q: Can I simplify expressions with radicals?

A: Yes, you can simplify expressions with radicals by applying the rule for radicals, which states that for any non-zero number $a$ and any integer $n$, $\sqrt[n]{a^n} = a$.

Q: How do I simplify expressions with absolute values?

A: To simplify expressions with absolute values, you need to apply the rule for absolute values, which states that for any non-zero number $a$, $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$.

Q: Can I simplify expressions with complex numbers?

A: Yes, you can simplify expressions with complex numbers by applying the rule for complex numbers, which states that for any complex number $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, $a + bi = \sqrt{a^2 + b^2} \left(\cos \theta + i \sin \theta\right)$, where $\theta$ is the argument of the complex number.

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponents. We have covered topics such as the Power of a Product Rule, the Power of a Power Rule, combining like terms, and simplifying expressions with negative exponents, zero exponents, fractional exponents, multiple bases, radicals, absolute values, and complex numbers. By understanding and applying these rules, you can simplify complex expressions and solve problems with ease.

Additional Resources

• Exponents and Powers: A comprehensive guide to exponents and powers, including rules and examples.
• Simplifying Expressions: A tutorial on simplifying expressions with multiple exponents.
• Mathematics Tutorials: A collection of tutorials on various mathematics topics, including exponents and powers.

Final Thoughts

Simplifying expressions with exponents is an essential skill in mathematics. By understanding and applying the rules of exponents, you can simplify complex expressions and solve problems with ease. Remember to always apply the Power of a Product Rule and the Power of a Power Rule, and then combine like terms to get the final simplified expression.