Simplify ( 5 A 3 ) 2 − 2 ( 3 A 3 ) 2 \left(5 A^3\right)^2 - 2\left(3 A^3\right)^2 ( 5 A 3 ) 2 − 2 ( 3 A 3 ) 2 .

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Understanding the Problem

The given expression involves the simplification of a mathematical expression that includes exponents and powers. The expression is $\left(5 a^3\right)^2 - 2\left(3 a^3\right)^2$. To simplify this expression, we need to apply the rules of exponents and powers.

Applying the Rules of Exponents

The first step in simplifying the expression is to apply the rule of exponents that states $(a^m)^n = a^{mn}$. Using this rule, we can rewrite the expression as follows:

$\left(5 a^3\right)^2 = 5^2 \cdot (a^3)^2 = 25 \cdot a^6$

$\left(3 a^3\right)^2 = 3^2 \cdot (a^3)^2 = 9 \cdot a^6$

Simplifying the Expression

Now that we have rewritten the expression using the rule of exponents, we can simplify it further by substituting the rewritten expressions back into the original expression:

$\left(5 a^3\right)^2 - 2\left(3 a^3\right)^2 = 25 \cdot a^6 - 2 \cdot 9 \cdot a^6$

Combining Like Terms

The next step in simplifying the expression is to combine like terms. In this case, we have two terms that have the same base ($a^6$) and the same exponent (6). We can combine these terms by adding or subtracting their coefficients:

$25 \cdot a^6 - 2 \cdot 9 \cdot a^6 = (25 - 18) \cdot a^6$

Evaluating the Expression

Now that we have combined like terms, we can evaluate the expression by simplifying the coefficient:

$(25 - 18) \cdot a^6 = 7 \cdot a^6$

Conclusion

In conclusion, the simplified expression is $7 \cdot a^6$. This is the final answer to the problem.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Apply the rule of exponents to rewrite the expression: $\left(5 a^3\right)^2 = 25 \cdot a^6$ and $\left(3 a^3\right)^2 = 9 \cdot a^6$
2. Substitute the rewritten expressions back into the original expression: $\left(5 a^3\right)^2 - 2\left(3 a^3\right)^2 = 25 \cdot a^6 - 2 \cdot 9 \cdot a^6$
3. Combine like terms: $(25 - 18) \cdot a^6 = 7 \cdot a^6$
4. Evaluate the expression: $7 \cdot a^6$

Final Answer

The final answer to the problem is $\boxed{7 a^6}$.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions involving exponents and powers:

• Always apply the rule of exponents first.
• Use the distributive property to simplify expressions.
• Combine like terms to simplify expressions.
• Evaluate the expression by simplifying the coefficient.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions involving exponents and powers:

• Failing to apply the rule of exponents.
• Not using the distributive property to simplify expressions.
• Not combining like terms to simplify expressions.
• Not evaluating the expression by simplifying the coefficient.

Real-World Applications

Here are some real-world applications of simplifying expressions involving exponents and powers:

• Simplifying expressions in algebra and calculus.
• Solving equations and inequalities involving exponents and powers.
• Working with exponential functions and their graphs.
• Understanding the concept of growth and decay in finance and economics.

Conclusion

In conclusion, simplifying expressions involving exponents and powers is an important skill in mathematics. By applying the rules of exponents and powers, combining like terms, and evaluating the expression, we can simplify complex expressions and arrive at the final answer.

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Frequently Asked Questions

Q: What is the rule of exponents?

A: The rule of exponents states that $(a^m)^n = a^{mn}$. This means that when we raise a power to a power, we multiply the exponents.

Q: How do I apply the rule of exponents to simplify expressions?

A: To apply the rule of exponents, we need to identify the base and the exponents in the expression. We then multiply the exponents and simplify the expression.

Q: What is the distributive property?

A: The distributive property is a rule that allows us to multiply a single term by multiple terms. It states that $a(b + c) = ab + ac$.

Q: How do I use the distributive property to simplify expressions?

A: To use the distributive property, we need to identify the terms in the expression that we want to multiply. We then multiply each term by the other term and simplify the expression.

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

A: A coefficient is a number that is multiplied by a variable. A variable is a letter or symbol that represents a value.

Q: How do I combine like terms?

A: To combine like terms, we need to identify the terms in the expression that have the same variable and exponent. We then add or subtract the coefficients of the like terms.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{7 a^6}$.

Q: What are some real-world applications of simplifying expressions involving exponents and powers?

A: Some real-world applications of simplifying expressions involving exponents and powers include:

• Simplifying expressions in algebra and calculus.
• Solving equations and inequalities involving exponents and powers.
• Working with exponential functions and their graphs.
• Understanding the concept of growth and decay in finance and economics.

Q: What are some common mistakes to avoid when simplifying expressions involving exponents and powers?

A: Some common mistakes to avoid when simplifying expressions involving exponents and powers include:

• Failing to apply the rule of exponents.
• Not using the distributive property to simplify expressions.
• Not combining like terms to simplify expressions.
• Not evaluating the expression by simplifying the coefficient.

Q: How can I practice simplifying expressions involving exponents and powers?

A: You can practice simplifying expressions involving exponents and powers by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions on your own and checking your work with a calculator or online tool.

Additional Resources

Here are some additional resources to help you learn more about simplifying expressions involving exponents and powers:

• Khan Academy: Exponents and Powers
• Mathway: Exponents and Powers
• Wolfram Alpha: Exponents and Powers
• MIT OpenCourseWare: Exponents and Powers

Conclusion

In conclusion, simplifying expressions involving exponents and powers is an important skill in mathematics. By applying the rules of exponents and powers, combining like terms, and evaluating the expression, we can simplify complex expressions and arrive at the final answer. We hope this Q&A article has been helpful in answering your questions and providing additional resources to help you learn more about simplifying expressions involving exponents and powers.