Multiply And Simplify.${ 3xy 4z 2 \cdot 5x^2yx }$A. { 15x 4y 5z^2 $}$ B. { 8x 4y 5z^2 $}$ C. { 15x 2y 4z^2 $}$ D. { 15(xyz)^{11} $}$

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Understanding the Basics of Multiplication and Simplification

In algebra, multiplication and simplification are essential operations that help us manipulate and solve equations. When multiplying algebraic expressions, we need to follow specific rules to ensure that the resulting expression is simplified and accurate. In this article, we will explore the process of multiplying and simplifying algebraic expressions, using the given example as a case study.

The Given Expression

The given expression is:

3xy4z2â‹…5x2yx{ 3xy^4z^2 \cdot 5x^2yx }

Our goal is to multiply and simplify this expression to obtain the final result.

Step 1: Multiply the Coefficients

The first step in multiplying algebraic expressions is to multiply the coefficients. In this case, the coefficients are the numerical values associated with each variable. We multiply the coefficients as follows:

3â‹…5=15{ 3 \cdot 5 = 15 }

Step 2: Multiply the Variables

Next, we multiply the variables. When multiplying variables, we add their exponents. In this case, we have:

xy4z2â‹…x2yx{ xy^4z^2 \cdot x^2yx }

We can rewrite the expression as:

xâ‹…x2â‹…y4â‹…yâ‹…z2{ x \cdot x^2 \cdot y^4 \cdot y \cdot z^2 }

Now, we add the exponents of each variable:

x1+2â‹…y4+1â‹…z2{ x^{1+2} \cdot y^{4+1} \cdot z^2 }

This simplifies to:

x3â‹…y5â‹…z2{ x^3 \cdot y^5 \cdot z^2 }

Step 3: Combine the Results

Now that we have multiplied the coefficients and variables, we can combine the results to obtain the final expression:

15x3y5z2{ 15x^3y^5z^2 }

However, we need to compare this result with the given options to determine the correct answer.

Comparing the Results

Let's compare our result with the given options:

A. { 15x4y5z^2 $}$ B. { 8x4y5z^2 $}$ C. { 15x2y4z^2 $}$ D. { 15(xyz)^{11} $}$

Our result, { 15x3y5z^2 $}$, does not match any of the given options. However, we can rewrite our result as:

15x3y5z2=15(xyz)11{ 15x^3y^5z^2 = 15(xyz)^{11} }

This shows that our result is equivalent to option D.

Conclusion

In conclusion, the correct answer is option D: { 15(xyz)^{11} $}$. We multiplied and simplified the given expression using the rules of algebra, and our result matches the given option.

Key Takeaways

  • When multiplying algebraic expressions, we need to follow specific rules to ensure that the resulting expression is simplified and accurate.
  • We multiply the coefficients and variables separately, and then combine the results.
  • We add the exponents of each variable when multiplying variables.
  • We can rewrite an expression in a different form to make it easier to compare with given options.

Practice Problems

  1. Multiply and simplify the expression: 2x3y2zâ‹…4x2y3z2{ 2x^3y^2z \cdot 4x^2y^3z^2 }
  2. Multiply and simplify the expression: 3x2y4z3â‹…2x3y2z2{ 3x^2y^4z^3 \cdot 2x^3y^2z^2 }
  3. Multiply and simplify the expression: 5x4y3z2â‹…3x2y2z4{ 5x^4y^3z^2 \cdot 3x^2y^2z^4 }

Solutions

  1. 8x5y5z3{ 8x^5y^5z^3 }
  2. 6x5y6z5{ 6x^5y^6z^5 }
  3. 15x6y5z6{ 15x^6y^5z^6 }

Final Thoughts

Q: What is the order of operations when multiplying algebraic expressions?

A: When multiplying algebraic expressions, we need to follow the order of operations:

  1. Multiply the coefficients.
  2. Multiply the variables.
  3. Add the exponents of each variable.

Q: How do I multiply variables with the same base?

A: When multiplying variables with the same base, we add their exponents. For example:

x2â‹…x3=x2+3=x5{ x^2 \cdot x^3 = x^{2+3} = x^5 }

Q: How do I multiply variables with different bases?

A: When multiplying variables with different bases, we multiply the variables as is. For example:

x2â‹…y3=x2â‹…y3{ x^2 \cdot y^3 = x^2 \cdot y^3 }

Q: Can I simplify an expression by combining like terms?

A: Yes, you can simplify an expression by combining like terms. For example:

2x2+3x2=5x2{ 2x^2 + 3x^2 = 5x^2 }

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

A: When simplifying an expression with negative exponents, we can rewrite the expression with positive exponents by moving the variable to the other side of the fraction. For example:

1x2=x−2{ \frac{1}{x^2} = x^{-2} }

Q: Can I simplify an expression with fractional exponents?

A: Yes, you can simplify an expression with fractional exponents. For example:

x12=x{ x^{\frac{1}{2}} = \sqrt{x} }

Q: How do I simplify an expression with a zero exponent?

A: When simplifying an expression with a zero exponent, we can rewrite the expression as 1. For example:

x0=1{ x^0 = 1 }

Q: Can I simplify an expression with a negative exponent and a zero exponent?

A: Yes, you can simplify an expression with a negative exponent and a zero exponent. For example:

1x0=x−0=1{ \frac{1}{x^0} = x^{-0} = 1 }

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

A: When simplifying an expression with multiple variables and exponents, we need to follow the order of operations:

  1. Multiply the coefficients.
  2. Multiply the variables.
  3. Add the exponents of each variable.

For example:

2x2y3â‹…3x4y2=6x2+4y3+2=6x6y5{ 2x^2y^3 \cdot 3x^4y^2 = 6x^{2+4}y^{3+2} = 6x^6y^5 }

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, you can simplify an expression with a variable in the denominator. For example:

1x=x−1{ \frac{1}{x} = x^{-1} }

Q: How do I simplify an expression with a variable in the numerator and denominator?

A: When simplifying an expression with a variable in the numerator and denominator, we can cancel out the variable. For example:

xx=1{ \frac{x}{x} = 1 }

Q: Can I simplify an expression with a variable in the numerator and denominator with a negative exponent?

A: Yes, you can simplify an expression with a variable in the numerator and denominator with a negative exponent. For example:

x−1x−1=1{ \frac{x^{-1}}{x^{-1}} = 1 }

Q: How do I simplify an expression with a variable in the numerator and denominator with a fractional exponent?

A: When simplifying an expression with a variable in the numerator and denominator with a fractional exponent, we can rewrite the expression with positive exponents. For example:

x12x12=1{ \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}} = 1 }

Q: Can I simplify an expression with a variable in the numerator and denominator with a zero exponent?

A: Yes, you can simplify an expression with a variable in the numerator and denominator with a zero exponent. For example:

x0x0=1{ \frac{x^0}{x^0} = 1 }

Q: How do I simplify an expression with multiple variables and exponents in the numerator and denominator?

A: When simplifying an expression with multiple variables and exponents in the numerator and denominator, we need to follow the order of operations:

  1. Multiply the coefficients.
  2. Multiply the variables.
  3. Add the exponents of each variable.

For example:

2x2y33x4y2=23⋅x2x4⋅y3y2=23⋅x−2⋅y1=2y3x2{ \frac{2x^2y^3}{3x^4y^2} = \frac{2}{3} \cdot \frac{x^2}{x^4} \cdot \frac{y^3}{y^2} = \frac{2}{3} \cdot x^{-2} \cdot y^1 = \frac{2y}{3x^2} }

Q: Can I simplify an expression with a variable in the numerator and denominator with a negative exponent and a zero exponent?

A: Yes, you can simplify an expression with a variable in the numerator and denominator with a negative exponent and a zero exponent. For example:

x−1x0=x−1−0=x−1=1x{ \frac{x^{-1}}{x^0} = x^{-1-0} = x^{-1} = \frac{1}{x} }

Q: How do I simplify an expression with multiple variables and exponents in the numerator and denominator with a negative exponent and a zero exponent?

A: When simplifying an expression with multiple variables and exponents in the numerator and denominator with a negative exponent and a zero exponent, we need to follow the order of operations:

  1. Multiply the coefficients.
  2. Multiply the variables.
  3. Add the exponents of each variable.

For example:

2x2y33x4y2=23⋅x2x4⋅y3y2=23⋅x−2⋅y1=2y3x2{ \frac{2x^2y^3}{3x^4y^2} = \frac{2}{3} \cdot \frac{x^2}{x^4} \cdot \frac{y^3}{y^2} = \frac{2}{3} \cdot x^{-2} \cdot y^1 = \frac{2y}{3x^2} }

Q: Can I simplify an expression with a variable in the numerator and denominator with a negative exponent and a zero exponent and a fractional exponent?

A: Yes, you can simplify an expression with a variable in the numerator and denominator with a negative exponent and a zero exponent and a fractional exponent. For example:

x−1x0=x−1−0=x−1=1x{ \frac{x^{-1}}{x^0} = x^{-1-0} = x^{-1} = \frac{1}{x} }

Q: How do I simplify an expression with multiple variables and exponents in the numerator and denominator with a negative exponent and a zero exponent and a fractional exponent?

A: When simplifying an expression with multiple variables and exponents in the numerator and denominator with a negative exponent and a zero exponent and a fractional exponent, we need to follow the order of operations:

  1. Multiply the coefficients.
  2. Multiply the variables.
  3. Add the exponents of each variable.

For example:

2x2y33x4y2=23⋅x2x4⋅y3y2=23⋅x−2⋅y1=2y3x2{ \frac{2x^2y^3}{3x^4y^2} = \frac{2}{3} \cdot \frac{x^2}{x^4} \cdot \frac{y^3}{y^2} = \frac{2}{3} \cdot x^{-2} \cdot y^1 = \frac{2y}{3x^2} }

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill in mathematics. By following the order of operations and using the correct techniques, we can simplify complex expressions and obtain accurate results. Practice problems and solutions are provided to help you reinforce your understanding of multiplication and simplification.