Simplify The Expression:$ \left(\frac{10 X^3 Y^2}{5 X^{-8} Y 4}\right) {-3}, \quad X \neq 0, \quad Y \neq 0 }$Choose The Correct Simplification A. { \frac{2 X^{18 }{y^8}$}$B. { \frac{x^9}{8 Y^5}$} C . \[ C. \[ C . \[ \frac{2

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Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for any math enthusiast. In this article, we will delve into the world of exponential expressions and explore the steps involved in simplifying them. We will use the given expression as a case study and apply the rules of exponents to simplify it.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, x3x^3 can be read as "x to the power of 3" and is equivalent to x×x×xx \times x \times x. Exponents can be positive, negative, or zero, and they can be applied to any number, variable, or expression.

The Given Expression

The given expression is (10x3y25x−8y4)−3\left(\frac{10 x^3 y^2}{5 x^{-8} y^4}\right)^{-3}. This expression involves fractions, exponents, and variables. Our goal is to simplify this expression using the rules of exponents.

Step 1: Simplify the Fraction

To simplify the fraction, we can start by canceling out any common factors in the numerator and denominator. In this case, we can cancel out the 5 in the numerator and denominator.

\frac{10 x^3 y^2}{5 x^{-8} y^4} = \frac{2 x^3 y^2}{x^{-8} y^4}

Step 2: Apply the Quotient Rule

The quotient rule states that when dividing two exponential expressions with the same base, we can subtract the exponents. In this case, we can apply the quotient rule to simplify the expression further.

\frac{2 x^3 y^2}{x^{-8} y^4} = 2 x^{3-(-8)} y^{2-4} = 2 x^{11} y^{-2}

Step 3: Apply the Power Rule

The power rule states that when raising an exponential expression to a power, we can multiply the exponents. In this case, we can apply the power rule to simplify the expression further.

(2 x^{11} y^{-2})^{-3} = 2^{-3} x^{-3 \times 11} y^{-3 \times (-2)} = \frac{2^{-3}}{x^{33} y^6}

Step 4: Simplify the Negative Exponent

A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. In this case, we can simplify the negative exponent by taking the reciprocal of the base.

\frac{2^{-3}}{x^{33} y^6} = \frac{1}{2^3 x^{33} y^6} = \frac{1}{8 x^{33} y^6}

Step 5: Simplify the Expression Further

We can simplify the expression further by applying the rule of exponents that states x−a=1xax^{-a} = \frac{1}{x^a}. In this case, we can rewrite the expression as follows:

\frac{1}{8 x^{33} y^6} = \frac{1}{8} x^{-33} y^{-6} = \frac{1}{8} \frac{1}{x^{33} y^6}

Conclusion

In conclusion, we have simplified the given expression using the rules of exponents. We started by simplifying the fraction, applying the quotient rule, and then applying the power rule. Finally, we simplified the negative exponent and the expression further. The simplified expression is 18x33y6\frac{1}{8 x^{33} y^6}.

Discussion

The correct simplification of the given expression is 18x33y6\frac{1}{8 x^{33} y^6}. This expression can be rewritten as x98y5\frac{x^9}{8 y^5} by applying the rule of exponents that states x−a=1xax^{-a} = \frac{1}{x^a}. Therefore, the correct answer is:

A. 2x18y8\frac{2 x^{18}}{y^8}

This option is incorrect because it does not match the simplified expression.

B. x98y5\frac{x^9}{8 y^5}

This option is correct because it matches the simplified expression.

C. 2x18y8\frac{2 x^{18}}{y^8}

This option is incorrect because it does not match the simplified expression.

In conclusion, the correct simplification of the given expression is x98y5\frac{x^9}{8 y^5}.

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Introduction

In our previous article, we explored the steps involved in simplifying exponential expressions. We used the given expression as a case study and applied the rules of exponents to simplify it. In this article, we will provide a Q&A guide to help you better understand the concepts and rules involved in simplifying exponential expressions.

Q&A

Q: What is the first step in simplifying an exponential expression?

A: The first step in simplifying an exponential expression is to simplify the fraction, if any, by canceling out any common factors in the numerator and denominator.

Q: What is the quotient rule in exponents?

A: The quotient rule states that when dividing two exponential expressions with the same base, we can subtract the exponents. For example, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.

Q: What is the power rule in exponents?

A: The power rule states that when raising an exponential expression to a power, we can multiply the exponents. For example, (xa)b=xab(x^a)^b = x^{ab}.

Q: How do we simplify a negative exponent?

A: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, x−a=1xax^{-a} = \frac{1}{x^a}.

Q: What is the rule for multiplying exponential expressions with the same base?

A: When multiplying exponential expressions with the same base, we can add the exponents. For example, xa×xb=xa+bx^a \times x^b = x^{a+b}.

Q: What is the rule for dividing exponential expressions with the same base?

A: When dividing exponential expressions with the same base, we can subtract the exponents. For example, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.

Q: How do we simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, we can apply the rules of exponents in the following order:

  1. Simplify the fraction, if any.
  2. Apply the quotient rule.
  3. Apply the power rule.
  4. Simplify the negative exponent, if any.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is x98y5\frac{x^9}{8 y^5}.

Examples

Example 1:

Simplify the expression: (3x2y32x−4y−2)−2\left(\frac{3 x^2 y^3}{2 x^{-4} y^{-2}}\right)^{-2}

Solution:

  1. Simplify the fraction: 3x2y32x−4y−2=3x2y32x4y2\frac{3 x^2 y^3}{2 x^{-4} y^{-2}} = \frac{3 x^2 y^3}{2} x^4 y^2
  2. Apply the quotient rule: 3x2y32x4y2=3x6y52\frac{3 x^2 y^3}{2} x^4 y^2 = \frac{3 x^6 y^5}{2}
  3. Apply the power rule: (3x6y52)−2=2232x12y10=49x12y10\left(\frac{3 x^6 y^5}{2}\right)^{-2} = \frac{2^2}{3^2 x^{12} y^{10}} = \frac{4}{9 x^{12} y^{10}}

Example 2:

Simplify the expression: (2x3y23x−5y−4)−3\left(\frac{2 x^3 y^2}{3 x^{-5} y^{-4}}\right)^{-3}

Solution:

  1. Simplify the fraction: 2x3y23x−5y−4=2x3y23x5y4\frac{2 x^3 y^2}{3 x^{-5} y^{-4}} = \frac{2 x^3 y^2}{3} x^5 y^4
  2. Apply the quotient rule: 2x3y23x5y4=2x8y63\frac{2 x^3 y^2}{3} x^5 y^4 = \frac{2 x^8 y^6}{3}
  3. Apply the power rule: (2x8y63)−3=3323x24y18=278x24y18\left(\frac{2 x^8 y^6}{3}\right)^{-3} = \frac{3^3}{2^3 x^{24} y^{18}} = \frac{27}{8 x^{24} y^{18}}

Conclusion

In conclusion, simplifying exponential expressions involves applying the rules of exponents in a specific order. By following these rules, you can simplify complex expressions and arrive at the final answer. Remember to simplify the fraction, apply the quotient rule, apply the power rule, and simplify the negative exponent, if any. With practice and patience, you will become proficient in simplifying exponential expressions.