Simplify The Expression \left(y^6 Z^9\right)\left(6 Y^4 Z^2\right ]. Write The Variables In Alphabetical Order. □ \square □

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Introduction

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In this article, we will simplify the given expression $\left(y^6 z^9\right)\left(6 y^4 z^2\right)$ and write the variables in alphabetical order. This involves applying the rules of exponents and following the correct order of operations.

Understanding the Expression

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The given expression is a product of two terms, each containing variables raised to certain powers. The first term is $y^6 z^9$, and the second term is $6 y^4 z^2$. To simplify this expression, we need to apply the rules of exponents, which state that when multiplying two terms with the same base, we add their exponents.

Applying the Rules of Exponents

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To simplify the expression, we will first multiply the coefficients (the numbers in front of the variables) and then add the exponents of the variables.

Multiplying the Coefficients

The coefficient of the first term is 1 (since there is no number in front of the variables), and the coefficient of the second term is 6. When multiplying these coefficients, we get:

1 × 6 = 6

Adding the Exponents of the Variables

The first term has $y^6$ and $z^9$, and the second term has $y^4$ and $z^2$. When adding the exponents of the variables, we get:

$y^6 \times y^4 = y^{6+4} = y^{10}$

$z^9 \times z^2 = z^{9+2} = z^{11}$

Writing the Variables in Alphabetical Order

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Now that we have simplified the expression, we need to write the variables in alphabetical order. The variables are $y$ and $z$, so we will write them in the order $y, z$.

Final Expression

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The final expression is:

$6y^{10}z^{11}$

Conclusion

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In this article, we simplified the given expression $\left(y^6 z^9\right)\left(6 y^4 z^2\right)$ and wrote the variables in alphabetical order. We applied the rules of exponents and followed the correct order of operations to get the final expression $6y^{10}z^{11}$.

Frequently Asked Questions

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Q: What are the rules of exponents?

A: The rules of exponents state that when multiplying two terms with the same base, we add their exponents. For example, $x^a \times x^b = x^{a+b}$.

Q: How do we add exponents?

A: To add exponents, we simply add the numbers in front of the variables. For example, $x^a \times x^b = x^{a+b}$.

Q: What is the correct order of operations?

A: The correct order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate expressions with exponents next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

Tips and Tricks

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Tip 1: Use the Rules of Exponents to Simplify Expressions

When simplifying expressions, use the rules of exponents to combine like terms.

Tip 2: Follow the Correct Order of Operations

To avoid errors, follow the correct order of operations: parentheses, exponents, multiplication and division, and addition and subtraction.

Tip 3: Write Variables in Alphabetical Order

When writing variables in alphabetical order, make sure to list them in the correct order (e.g., $y, z$).

Further Reading

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For more information on simplifying expressions and applying the rules of exponents, check out the following resources:

• Mathway: A online math problem solver that can help you simplify expressions and solve equations.
• Khan Academy: A free online learning platform that offers video lessons and practice exercises on math and other subjects.
• Wolfram Alpha: A powerful online calculator that can help you simplify expressions and solve equations.

References

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• Algebra: A branch of mathematics that deals with the study of variables and their relationships.
• Exponents: A mathematical operation that involves raising a number to a power.
• Order of Operations: A set of rules that dictate the order in which mathematical operations should be performed.
  

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Introduction

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In our previous article, we simplified the expression $\left(y^6 z^9\right)\left(6 y^4 z^2\right)$ and wrote the variables in alphabetical order. In this article, we will answer some frequently asked questions related to simplifying expressions and applying the rules of exponents.

Q&A

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Q: What are the rules of exponents?

A: The rules of exponents state that when multiplying two terms with the same base, we add their exponents. For example, $x^a \times x^b = x^{a+b}$.

Q: How do we add exponents?

A: To add exponents, we simply add the numbers in front of the variables. For example, $x^a \times x^b = x^{a+b}$.

Q: What is the correct order of operations?

A: The correct order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate expressions with exponents next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

Q: How do we simplify expressions with multiple variables?

A: To simplify expressions with multiple variables, we need to apply the rules of exponents and follow the correct order of operations. For example, if we have the expression $x^3y^2z^4$, we can simplify it by adding the exponents of the variables: $x^3y^2z^4 = x^{3+0}y^{2+0}z^{4+0} = x^3y^2z^4$.

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

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value. For example, in the expression $3x^2$, the number 3 is the coefficient and the letter x is the variable.

Q: How do we write variables in alphabetical order?

A: To write variables in alphabetical order, we simply list them in the correct order (e.g., $y, z$).

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

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

• Not following the correct order of operations
• Not applying the rules of exponents correctly
• Not writing variables in alphabetical order
• Not simplifying expressions with multiple variables

Tips and Tricks

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Tip 1: Use the Rules of Exponents to Simplify Expressions

When simplifying expressions, use the rules of exponents to combine like terms.

Tip 2: Follow the Correct Order of Operations

To avoid errors, follow the correct order of operations: parentheses, exponents, multiplication and division, and addition and subtraction.

Tip 3: Write Variables in Alphabetical Order

When writing variables in alphabetical order, make sure to list them in the correct order (e.g., $y, z$).

Further Reading

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For more information on simplifying expressions and applying the rules of exponents, check out the following resources:

• Mathway: A online math problem solver that can help you simplify expressions and solve equations.
• Khan Academy: A free online learning platform that offers video lessons and practice exercises on math and other subjects.
• Wolfram Alpha: A powerful online calculator that can help you simplify expressions and solve equations.

References

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• Algebra: A branch of mathematics that deals with the study of variables and their relationships.
• Exponents: A mathematical operation that involves raising a number to a power.
• Order of Operations: A set of rules that dictate the order in which mathematical operations should be performed.

Conclusion

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In this article, we answered some frequently asked questions related to simplifying expressions and applying the rules of exponents. We also provided some tips and tricks for simplifying expressions and avoiding common mistakes. By following the correct order of operations and applying the rules of exponents, you can simplify expressions and solve equations with confidence.