Which Expression Is Equivalent To $\frac{60 X^{20} Y^{24}}{30 X^{10} Y^{12}}$?A. $2 X^2 Y^2$B. $2 X^{10} Y^{12}$C. $30 X^2 Y^2$D. $30 X^{10} Y^{12}$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves reducing complex expressions to their simplest form, making it easier to solve equations and inequalities. In this article, we will focus on simplifying the given expression $\frac{60 x^{20} y^{24}}{30 x^{10} y^{12}}$ and explore the different options provided.

Understanding the Expression

The given expression is a fraction with two terms in the numerator and one term in the denominator. The numerator has two variables, x and y, raised to the power of 20 and 24, respectively. The denominator has one variable, x, raised to the power of 10, and one variable, y, raised to the power of 12.

Simplifying the Expression

To simplify the expression, we need to apply the rules of exponents. When dividing two terms with the same base, we subtract the exponents. In this case, we have:

60x20y2430x10y12=6030â‹…x20x10â‹…y24y12\frac{60 x^{20} y^{24}}{30 x^{10} y^{12}} = \frac{60}{30} \cdot \frac{x^{20}}{x^{10}} \cdot \frac{y^{24}}{y^{12}}

Applying the Rules of Exponents

Now, let's apply the rules of exponents to simplify the expression further:

  • When dividing two terms with the same base, we subtract the exponents. In this case, we have: $\frac{x{20}}{x{10}} = x^{20-10} = x^{10}$
  • Similarly, we have: $\frac{y{24}}{y{12}} = y^{24-12} = y^{12}$

Simplifying the Fraction

Now that we have simplified the terms with the same base, we can simplify the fraction:

6030=2\frac{60}{30} = 2

Combining the Terms

Now, let's combine the simplified terms:

2â‹…x10â‹…y122 \cdot x^{10} \cdot y^{12}

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options provided:

  • Option A: 2x2y22 x^2 y^2
  • Option B: 2x10y122 x^{10} y^{12}
  • Option C: 30x2y230 x^2 y^2
  • Option D: 30x10y1230 x^{10} y^{12}

Conclusion

Based on our simplification, we can see that the correct answer is:

  • Option B: 2x10y122 x^{10} y^{12}

This is because our simplified expression matches the option exactly.

Final Answer

The final answer is Option B: 2x10y122 x^{10} y^{12}.

Frequently Asked Questions

Q: What is the rule for dividing two terms with the same base?

A: When dividing two terms with the same base, we subtract the exponents.

Q: How do we simplify the fraction?

A: We simplify the fraction by dividing the numerator and denominator by their greatest common divisor.

Q: What is the final answer?

A: The final answer is Option B: 2x10y122 x^{10} y^{12}.

References

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving equations and inequalities. In this article, we will provide a comprehensive Q&A guide on algebraic expressions, covering various topics and concepts.

Q&A

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It is a way to represent a value or a relationship between values using mathematical symbols.

Q: What are the basic components of an algebraic expression?

A: The basic components of an algebraic expression are:

  • Variables: Letters or symbols that represent values or quantities.
  • Constants: Numbers that do not change value.
  • Mathematical operations: Addition, subtraction, multiplication, and division.

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

A: A variable is a letter or symbol that represents a value or quantity that can change, while a constant is a number that does not change value.

Q: How do we simplify algebraic expressions?

A: We simplify algebraic expressions by applying the rules of exponents, combining like terms, and rearranging the expression to make it easier to solve.

Q: What is the rule for dividing two terms with the same base?

A: When dividing two terms with the same base, we subtract the exponents.

Q: How do we evaluate an algebraic expression?

A: We evaluate an algebraic expression by substituting the values of the variables and performing the mathematical operations.

Q: What is the order of operations in algebraic expressions?

A: The order of operations in algebraic expressions is:

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

Q: How do we solve equations with algebraic expressions?

A: We solve equations with algebraic expressions by isolating the variable and using inverse operations to solve for the value of the variable.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do we graph algebraic expressions?

A: We graph algebraic expressions by plotting points on a coordinate plane and using the graph to visualize the relationship between the variables.

Conclusion

Algebraic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving equations and inequalities. By following the rules of exponents, combining like terms, and rearranging the expression, we can simplify algebraic expressions and solve equations. We hope this Q&A guide has provided a comprehensive overview of algebraic expressions and has helped you to better understand this important mathematical concept.

Frequently Asked Questions

Q: What is the difference between an algebraic expression and an equation?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations, while an equation is a statement that two expressions are equal.

Q: How do we simplify a rational expression?

A: We simplify a rational expression by canceling out any common factors in the numerator and denominator.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality in which the highest power of the variable is 1, while a quadratic inequality is an inequality in which the highest power of the variable is 2.

Q: How do we solve a system of equations with algebraic expressions?

A: We solve a system of equations with algebraic expressions by using substitution or elimination methods to find the values of the variables.

References

Further Reading

Related Topics