Find The Quotient Of $\left(2 A^5 B^3\right)^2 \cdot 12 A B^4$ And $5 A^7 B^3 + 3 A^7 B^3$.

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Introduction

In algebra, finding the quotient of two expressions involves dividing one expression by another. This process can be complex, especially when dealing with variables and exponents. In this article, we will explore how to find the quotient of two given algebraic expressions.

The Given Expressions

The two expressions we need to find the quotient of are:

(2a5b3)2â‹…12ab4\left(2 a^5 b^3\right)^2 \cdot 12 a b^4 and 5a7b3+3a7b35 a^7 b^3 + 3 a^7 b^3.

Simplifying the First Expression

To simplify the first expression, we need to expand the squared term using the exponent rule (ab)n=anbn\left(ab\right)^n = a^n b^n. This gives us:

(2a5b3)2=4a10b6\left(2 a^5 b^3\right)^2 = 4 a^{10} b^6

Now, we multiply this result by 12ab412 a b^4:

4a10b6â‹…12ab4=48a11b104 a^{10} b^6 \cdot 12 a b^4 = 48 a^{11} b^{10}

Simplifying the Second Expression

The second expression is a sum of two terms:

5a7b3+3a7b35 a^7 b^3 + 3 a^7 b^3

We can factor out the common term a7b3a^7 b^3:

a7b3(5+3)=8a7b3a^7 b^3 (5 + 3) = 8 a^7 b^3

Finding the Quotient

Now that we have simplified both expressions, we can find the quotient by dividing the first expression by the second expression:

48a11b108a7b3\frac{48 a^{11} b^{10}}{8 a^7 b^3}

Applying the Quotient Rule

To simplify the quotient, we can apply the quotient rule, which states that when dividing like bases, we subtract the exponents:

48a11b108a7b3=6a11−7b10−3\frac{48 a^{11} b^{10}}{8 a^7 b^3} = 6 a^{11-7} b^{10-3}

Simplifying the Quotient

Now, we can simplify the quotient by evaluating the exponents:

6a4b76 a^{4} b^7

Conclusion

In this article, we found the quotient of two algebraic expressions by simplifying each expression and then dividing them. We applied the quotient rule to simplify the result and obtained the final quotient.

Final Answer

The final answer is 6a4b7\boxed{6 a^{4} b^7}.

Step-by-Step Solution

Here is the step-by-step solution to find the quotient:

  1. Simplify the first expression by expanding the squared term and multiplying by the second term.
  2. Simplify the second expression by factoring out the common term.
  3. Find the quotient by dividing the first expression by the second expression.
  4. Apply the quotient rule to simplify the result.
  5. Simplify the quotient by evaluating the exponents.

Frequently Asked Questions

Q: What is the quotient of two algebraic expressions?

A: The quotient of two algebraic expressions is the result of dividing one expression by another.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can use the exponent rules, such as expanding squared terms and factoring out common terms.

Q: What is the quotient rule?

A: The quotient rule states that when dividing like bases, you subtract the exponents.

Q: How do I find the quotient of two expressions?

A: To find the quotient of two expressions, you can follow these steps:

  1. Simplify each expression.
  2. Divide the first expression by the second expression.
  3. Apply the quotient rule to simplify the result.
  4. Simplify the quotient by evaluating the exponents.

Related Topics

  • Simplifying algebraic expressions
  • Applying the quotient rule
  • Finding the quotient of two expressions

References

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Introduction

In our previous article, we explored how to find the quotient of two algebraic expressions. However, we understand that there may be many questions and doubts that readers may have. In this article, we will address some of the most frequently asked questions related to finding the quotient of algebraic expressions.

Q&A

Q: What is the quotient of two algebraic expressions?

A: The quotient of two algebraic expressions is the result of dividing one expression by another.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can use the exponent rules, such as expanding squared terms and factoring out common terms.

Q: What is the quotient rule?

A: The quotient rule states that when dividing like bases, you subtract the exponents.

Q: How do I find the quotient of two expressions?

A: To find the quotient of two expressions, you can follow these steps:

  1. Simplify each expression.
  2. Divide the first expression by the second expression.
  3. Apply the quotient rule to simplify the result.
  4. Simplify the quotient by evaluating the exponents.

Q: What are some common mistakes to avoid when finding the quotient of algebraic expressions?

A: Some common mistakes to avoid include:

  • Not simplifying the expressions before dividing
  • Not applying the quotient rule correctly
  • Not evaluating the exponents correctly

Q: How do I handle negative exponents when finding the quotient of algebraic expressions?

A: When handling negative exponents, you can rewrite the expression with a positive exponent by moving the base to the other side of the fraction bar.

Q: Can I use a calculator to find the quotient of algebraic expressions?

A: While calculators can be useful for simplifying expressions, it's generally recommended to use algebraic methods to find the quotient of algebraic expressions.

Q: How do I check my work when finding the quotient of algebraic expressions?

A: To check your work, you can multiply the quotient by the divisor to see if you get the dividend.

Q: What are some real-world applications of finding the quotient of algebraic expressions?

A: Finding the quotient of algebraic expressions has many real-world applications, including:

  • Calculating rates of change
  • Finding the area of shapes
  • Determining the volume of objects

Conclusion

Finding the quotient of algebraic expressions can be a challenging task, but with practice and patience, you can master it. Remember to simplify the expressions, apply the quotient rule, and evaluate the exponents correctly. If you have any further questions or doubts, feel free to ask.

Final Tips

  • Practice, practice, practice! The more you practice finding the quotient of algebraic expressions, the more confident you'll become.
  • Use algebraic methods to simplify expressions, rather than relying on calculators.
  • Check your work by multiplying the quotient by the divisor.
  • Apply the quotient rule correctly to simplify the result.

Related Topics

  • Simplifying algebraic expressions
  • Applying the quotient rule
  • Finding the quotient of two expressions

References

Additional Resources