Choose The Correct Simplification Of $\left(8 X^4 Y^3\right)^2$.A. $16 X^6 Y^5$B. $64 X^8 Y^6$C. $64 X^6 Y^5$D. $16 X^8 Y^6$

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Understanding Exponents and Simplification

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When simplifying expressions with exponents, it's essential to understand the rules governing exponentiation. In this article, we will explore the correct simplification of the expression (8x4y3)2\left(8 x^4 y^3\right)^2.

The Power of a Product Rule

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

(ab)n=anbn(ab)^n = a^n b^n

Using this rule, we can simplify the expression (8x4y3)2\left(8 x^4 y^3\right)^2 by applying the power of a product rule.

Simplifying the Expression

To simplify the expression (8x4y3)2\left(8 x^4 y^3\right)^2, we need to apply the power of a product rule. This means that each number in the product is raised to the power of 2.

(8x4y3)2=82(x4)2(y3)2\left(8 x^4 y^3\right)^2 = 8^2 \left(x^4\right)^2 \left(y^3\right)^2

Using the power of a product rule, we can rewrite the expression as:

82(x4)2(y3)2=64x4â‹…2y3â‹…28^2 \left(x^4\right)^2 \left(y^3\right)^2 = 64 x^{4 \cdot 2} y^{3 \cdot 2}

Applying Exponent Rules

Now that we have simplified the expression using the power of a product rule, we can apply exponent rules to further simplify the expression.

When multiplying numbers with the same base, we can add their exponents. In this case, we have:

x4â‹…2=x8x^{4 \cdot 2} = x^8

and

y3â‹…2=y6y^{3 \cdot 2} = y^6

Therefore, the simplified expression is:

64x8y664 x^8 y^6

Conclusion

In conclusion, the correct simplification of the expression (8x4y3)2\left(8 x^4 y^3\right)^2 is 64x8y664 x^8 y^6. This can be achieved by applying the power of a product rule and exponent rules.

Answer Key

The correct answer is:

  • B. 64x8y664 x^8 y^6

Additional Tips and Tricks

When simplifying expressions with exponents, it's essential to remember the following tips and tricks:

  • Power of a Product Rule: When a product of numbers is raised to a power, each number in the product is raised to that power.
  • Exponent Rules: When multiplying numbers with the same base, we can add their exponents.
  • Simplify Exponents: Simplify exponents by applying exponent rules and the power of a product rule.

By following these tips and tricks, you can simplify expressions with exponents like a pro!

Common Mistakes to Avoid

When simplifying expressions with exponents, it's essential to avoid the following common mistakes:

  • Incorrect Application of Exponent Rules: Make sure to apply exponent rules correctly when multiplying numbers with the same base.
  • Incorrect Simplification of Exponents: Make sure to simplify exponents correctly by applying exponent rules and the power of a product rule.
  • Incorrect Answer: Make sure to choose the correct answer from the options provided.

By avoiding these common mistakes, you can ensure that your simplifications are accurate and correct.

Real-World Applications

Simplifying expressions with exponents has numerous real-world applications in various fields, including:

  • Science: Exponents are used to represent repeated multiplication of numbers, which is essential in scientific calculations.
  • Engineering: Exponents are used to represent repeated multiplication of numbers, which is essential in engineering calculations.
  • Finance: Exponents are used to represent repeated multiplication of numbers, which is essential in financial calculations.

By understanding and applying exponent rules, you can simplify expressions with exponents and solve complex problems in various fields.

Conclusion

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about simplifying expressions with exponents.

Q: What is the power of a product rule?

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

(ab)n=anbn(ab)^n = a^n b^n

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the power of a product rule and exponent rules. Here's a step-by-step guide:

  1. Apply the power of a product rule by raising each number in the product to the power.
  2. Simplify exponents by applying exponent rules.
  3. Combine like terms.

Q: What are exponent rules?

A: Exponent rules are a set of rules that govern the behavior of exponents. Some of the most common exponent rules include:

  • Product Rule: When multiplying numbers with the same base, we can add their exponents.
  • Power Rule: When raising a power to a power, we can multiply the exponents.
  • Quotient Rule: When dividing numbers with the same base, we can subtract their exponents.

Q: How do I apply exponent rules?

A: To apply exponent rules, you need to follow these steps:

  1. Identify the base and the exponents.
  2. Apply the product rule by adding the exponents.
  3. Apply the power rule by multiplying the exponents.
  4. Apply the quotient rule by subtracting the exponents.

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:

  • Incorrect Application of Exponent Rules: Make sure to apply exponent rules correctly when multiplying numbers with the same base.
  • Incorrect Simplification of Exponents: Make sure to simplify exponents correctly by applying exponent rules and the power of a product rule.
  • Incorrect Answer: Make sure to choose the correct answer from the options provided.

Q: How do I choose the correct answer when simplifying expressions with exponents?

A: To choose the correct answer when simplifying expressions with exponents, you need to follow these steps:

  1. Simplify the expression using exponent rules and the power of a product rule.
  2. Compare the simplified expression with the answer options.
  3. Choose the answer that matches the simplified expression.

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

A: Simplifying expressions with exponents has numerous real-world applications in various fields, including:

  • Science: Exponents are used to represent repeated multiplication of numbers, which is essential in scientific calculations.
  • Engineering: Exponents are used to represent repeated multiplication of numbers, which is essential in engineering calculations.
  • Finance: Exponents are used to represent repeated multiplication of numbers, which is essential in financial calculations.

Conclusion

In conclusion, simplifying expressions with exponents is a crucial skill that has numerous real-world applications. By understanding and applying exponent rules, you can simplify expressions with exponents and solve complex problems in various fields. Remember to apply the power of a product rule and exponent rules to simplify exponents, and avoid common mistakes like incorrect application of exponent rules and incorrect simplification of exponents. With practice and patience, you can become a master of simplifying expressions with exponents!

Additional Resources

For more information on simplifying expressions with exponents, check out the following resources:

  • Math textbooks: Math textbooks provide a comprehensive guide to simplifying expressions with exponents.
  • Online resources: Online resources, such as Khan Academy and Mathway, provide interactive lessons and practice exercises on simplifying expressions with exponents.
  • Practice problems: Practice problems, such as those found on websites like IXL and Math Open Reference, provide opportunities to practice simplifying expressions with exponents.

By following these resources and practicing regularly, you can become proficient in simplifying expressions with exponents and solve complex problems in various fields.