Simplify $\frac{x^2\left(y^3\right)^4}{x Y^5} = X^a \cdot Y^b$.Find The Values Of $a$ And $b$:$\begin{array}{l} a = \\ b = \end{array}$

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Introduction

In algebra, simplifying complex expressions is a crucial skill that helps us solve equations and inequalities. One of the most common techniques used to simplify expressions is the application of the rules of exponents. In this article, we will simplify the given algebraic expression and find the values of a and b.

The Given Expression

The given expression is x2(y3)4xy5=xaâ‹…yb\frac{x^2\left(y^3\right)^4}{x y^5} = x^a \cdot y^b. Our goal is to simplify this expression and find the values of a and b.

Simplifying the Expression

To simplify the expression, we will start by applying the rules of exponents. The first step is to simplify the numerator. We can do this by applying the power rule, which states that (am)n=amn\left(a^m\right)^n = a^{mn}.

Using this rule, we can simplify the numerator as follows:

(y3)4=y3â‹…4=y12\left(y^3\right)^4 = y^{3 \cdot 4} = y^{12}

Now, we can rewrite the expression as:

x2â‹…y12xy5\frac{x^2 \cdot y^{12}}{x y^5}

Canceling Out Common Factors

The next step is to cancel out common factors in the numerator and denominator. We can do this by dividing both the numerator and denominator by the common factor, which is xx.

x2â‹…y12xy5=xâ‹…y12y5\frac{x^2 \cdot y^{12}}{x y^5} = \frac{x \cdot y^{12}}{y^5}

Applying the Quotient Rule

Now, we can apply the quotient rule, which states that aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Using this rule, we can simplify the expression as follows:

x⋅y12y5=x⋅y12−5=x⋅y7\frac{x \cdot y^{12}}{y^5} = x \cdot y^{12-5} = x \cdot y^7

Finding the Values of a and b

Now that we have simplified the expression, we can find the values of a and b. We can do this by comparing the simplified expression with the original expression.

xaâ‹…yb=xâ‹…y7x^a \cdot y^b = x \cdot y^7

Comparing the two expressions, we can see that:

a=1a = 1 b=7b = 7

Conclusion

In this article, we simplified the given algebraic expression and found the values of a and b. We applied the rules of exponents, canceled out common factors, and used the quotient rule to simplify the expression. We then compared the simplified expression with the original expression to find the values of a and b.

Frequently Asked Questions

  • What is the power rule of exponents? The power rule of exponents states that (am)n=amn\left(a^m\right)^n = a^{mn}.
  • What is the quotient rule of exponents? The quotient rule of exponents states that aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • How do I simplify an algebraic expression? To simplify an algebraic expression, you can apply the rules of exponents, cancel out common factors, and use the quotient rule.

Final Answer

The final answer is:

a=1,b=7\boxed{a = 1, b = 7}

Introduction

In our previous article, we simplified the algebraic expression x2(y3)4xy5=xaâ‹…yb\frac{x^2\left(y^3\right)^4}{x y^5} = x^a \cdot y^b and found the values of a and b. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to simplify algebraic expressions.

Q&A Guide

Q: What is the power rule of exponents?

A: The power rule of exponents states that (am)n=amn\left(a^m\right)^n = a^{mn}. This means that when you raise a power to another power, you multiply the exponents.

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that aman=am−n\frac{a^m}{a^n} = a^{m-n}. This means that when you divide two powers with the same base, you subtract the exponents.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can follow these steps:

  1. Apply the rules of exponents, such as the power rule and the quotient rule.
  2. Cancel out common factors in the numerator and denominator.
  3. Use the quotient rule to simplify the expression.

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

A: A variable is a letter or symbol that represents a value that can change, such as x or y. A constant is a value that does not change, such as 2 or 5.

Q: How do I apply the power rule of exponents?

A: To apply the power rule of exponents, you can follow these steps:

  1. Identify the base and the exponent in the expression.
  2. Raise the base to the power of the exponent.
  3. Multiply the exponents.

Q: How do I apply the quotient rule of exponents?

A: To apply the quotient rule of exponents, you can follow these steps:

  1. Identify the base and the exponents in the expression.
  2. Subtract the exponents.
  3. Write the result as a power of the base.

Q: What is the order of operations in algebra?

A: The order of operations in algebra 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: Evaluate addition and subtraction operations from left to right.

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

A: To simplify an expression with multiple variables, you can follow these steps:

  1. Identify the variables and their exponents in the expression.
  2. Apply the rules of exponents, such as the power rule and the quotient rule.
  3. Cancel out common factors in the numerator and denominator.
  4. Use the quotient rule to simplify the expression.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques used to simplify algebraic expressions. We covered topics such as the power rule of exponents, the quotient rule of exponents, and the order of operations in algebra. We also provided examples and step-by-step instructions to help you simplify expressions with multiple variables.

Frequently Asked Questions

  • What is the difference between a variable and a constant?
  • How do I apply the power rule of exponents?
  • How do I apply the quotient rule of exponents?
  • What is the order of operations in algebra?
  • How do I simplify an expression with multiple variables?

Final Answer

The final answer is:

  • The power rule of exponents states that (am)n=amn\left(a^m\right)^n = a^{mn}.
  • The quotient rule of exponents states that aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • The order of operations in algebra is: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
  • To simplify an expression with multiple variables, you can apply the rules of exponents, cancel out common factors, and use the quotient rule.