Simplify The Expression: ( − 3 Y 2 W 3 ) 4 ( 2 Y 6 ) 5 \left(-3 Y^2 W^3\right)^4\left(2 Y^6\right)^5 ( − 3 Y 2 W 3 ) 4 ( 2 Y 6 ) 5

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Understanding the Problem

When dealing with expressions involving exponents, it's essential to understand the rules of exponentiation. The given expression involves two terms, each raised to a power. We need to simplify this expression by applying the rules of exponentiation.

Rules of Exponentiation

Before we dive into simplifying the expression, let's review the rules of exponentiation:

• When a power is raised to another power, we multiply the exponents: $\left(a^m\right)^n = a^{m \cdot n}$
• When we multiply two powers with the same base, we add the exponents: $a^m \cdot a^n = a^{m + n}$
• When we divide two powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m - n}$

Simplifying the Expression

Now that we've reviewed the rules of exponentiation, let's simplify the given expression:

$\left(-3 y^2 w^3\right)^4\left(2 y^6\right)^5$

Using the first rule of exponentiation, we can rewrite the expression as:

$(-3)^4 \cdot (y^2)^4 \cdot (w^3)^4 \cdot 2^5 \cdot (y^6)^5$

Applying the Rules of Exponentiation

Now, let's apply the rules of exponentiation to simplify each term:

• $(-3)^4 = (-3) \cdot (-3) \cdot (-3) \cdot (-3) = 81$
• $(y^2)^4 = y^{2 \cdot 4} = y^8$
• $(w^3)^4 = w^{3 \cdot 4} = w^{12}$
• $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$
• $(y^6)^5 = y^{6 \cdot 5} = y^{30}$

Combining the Terms

Now that we've simplified each term, let's combine them:

$81 \cdot y^8 \cdot w^{12} \cdot 32 \cdot y^{30}$

Final Simplification

Finally, let's simplify the expression by combining like terms:

$81 \cdot 32 \cdot y^{8 + 30} \cdot w^{12}$

$= 2592 \cdot y^{38} \cdot w^{12}$

The final simplified expression is:

$2592 y^{38} w^{12}$

Conclusion

In this article, we simplified the expression $\left(-3 y^2 w^3\right)^4\left(2 y^6\right)^5$ by applying the rules of exponentiation. We reviewed the rules of exponentiation and then applied them to simplify each term in the expression. Finally, we combined the simplified terms to obtain the final simplified expression.

Frequently Asked Questions

• Q: What is the rule for raising a power to another power? A: When a power is raised to another power, we multiply the exponents: $\left(a^m\right)^n = a^{m \cdot n}$.
• Q: What is the rule for multiplying two powers with the same base? A: When we multiply two powers with the same base, we add the exponents: $a^m \cdot a^n = a^{m + n}$.
• Q: What is the rule for dividing two powers with the same base? A: When we divide two powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m - n}$.

Further Reading

If you're interested in learning more about exponentiation and simplifying expressions, here are some additional resources:

• Khan Academy: Exponent Rules
• Mathway: Exponent Rules
• Wolfram Alpha: Exponent Rules

References

• [1] Khan Academy. (n.d.). Exponent Rules. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponents/x2f-exponent-rules/x2f-exponent-rules/v/exponent-rules
• [2] Mathway. (n.d.). Exponent Rules. Retrieved from https://www.mathway.com/subjects/exponent-rules
• [3] Wolfram Alpha. (n.d.). Exponent Rules. Retrieved from https://www.wolframalpha.com/input/?i=exponent+rules
  

Introduction

Exponents can be a challenging concept to grasp, especially when it comes to simplifying expressions. However, with the right rules and techniques, you can simplify even the most complex expressions with ease. In this article, we'll answer some of the most frequently asked questions about exponent rules and provide you with the tools you need to simplify expressions like a pro.

Q: What is the rule for raising a power to another power?

A: When a power is raised to another power, we multiply the exponents: $\left(a^m\right)^n = a^{m \cdot n}$. This means that if we have an expression like $\left(2^3\right)^4$, we can simplify it by multiplying the exponents: $2^{3 \cdot 4} = 2^{12}$.

Q: What is the rule for multiplying two powers with the same base?

A: When we multiply two powers with the same base, we add the exponents: $a^m \cdot a^n = a^{m + n}$. This means that if we have an expression like $2^3 \cdot 2^4$, we can simplify it by adding the exponents: $2^{3 + 4} = 2^7$.

Q: What is the rule for dividing two powers with the same base?

A: When we divide two powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m - n}$. This means that if we have an expression like $\frac{2^5}{2^3}$, we can simplify it by subtracting the exponents: $2^{5 - 3} = 2^2$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, we need to apply the rules of exponentiation in the correct order. We start by simplifying the exponents inside the parentheses, and then we apply the rules of exponentiation to simplify the expression. For example, if we have the expression $\left(2^3\right)^4 \cdot 2^5$, we can simplify it by first simplifying the exponents inside the parentheses: $2^{3 \cdot 4} = 2^{12}$. Then, we can simplify the expression by adding the exponents: $2^{12} \cdot 2^5 = 2^{12 + 5} = 2^{17}$.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, we need to apply the rule of negative exponents: $a^{-m} = \frac{1}{a^m}$. This means that if we have an expression like $2^{-3}$, we can simplify it by rewriting it as a fraction: $\frac{1}{2^3}$. Then, we can simplify the expression by applying the rules of exponentiation: $\frac{1}{2^3} = \frac{1}{8}$.

Q: How do I simplify an expression with fractional exponents?

A: To simplify an expression with fractional exponents, we need to apply the rule of fractional exponents: $a^{m/n} = \sqrt[n]{a^m}$. This means that if we have an expression like $2^{3/4}$, we can simplify it by rewriting it as a radical: $\sqrt[4]{2^3}$. Then, we can simplify the expression by applying the rules of exponentiation: $\sqrt[4]{2^3} = \sqrt[4]{8} = 2$.

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

A: Some common mistakes to avoid when simplifying expressions with exponents include:

• Not applying the rules of exponentiation in the correct order
• Not simplifying the exponents inside the parentheses
• Not applying the rule of negative exponents
• Not applying the rule of fractional exponents
• Not simplifying the expression by adding or subtracting the exponents

Conclusion

Simplifying expressions with exponents can be a challenging task, but with the right rules and techniques, you can simplify even the most complex expressions with ease. By applying the rules of exponentiation and avoiding common mistakes, you can simplify expressions like a pro. Remember to always simplify the exponents inside the parentheses, apply the rules of exponentiation in the correct order, and simplify the expression by adding or subtracting the exponents.

Frequently Asked Questions

• Q: What is the rule for raising a power to another power? A: When a power is raised to another power, we multiply the exponents: $\left(a^m\right)^n = a^{m \cdot n}$.
• Q: What is the rule for multiplying two powers with the same base? A: When we multiply two powers with the same base, we add the exponents: $a^m \cdot a^n = a^{m + n}$.
• Q: What is the rule for dividing two powers with the same base? A: When we divide two powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m - n}$.

Further Reading

If you're interested in learning more about exponent rules and simplifying expressions, here are some additional resources:

• Khan Academy: Exponent Rules
• Mathway: Exponent Rules
• Wolfram Alpha: Exponent Rules

References

• [1] Khan Academy. (n.d.). Exponent Rules. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponents/x2f-exponent-rules/x2f-exponent-rules/v/exponent-rules
• [2] Mathway. (n.d.). Exponent Rules. Retrieved from https://www.mathway.com/subjects/exponent-rules
• [3] Wolfram Alpha. (n.d.). Exponent Rules. Retrieved from https://www.wolframalpha.com/input/?i=exponent+rules