Tyler Simplified The Expression X − 3 Y − 9 X^{-3} Y^{-9} X − 3 Y − 9 . His Procedure Is Shown Below. X − 3 Y − 9 = 1 X 3 ⋅ 1 Y − 9 = 1 X 3 Y − 9 X^{-3} Y^{-9} = \frac{1}{x^3} \cdot \frac{1}{y^{-9}} = \frac{1}{x^3 Y^{-9}} X − 3 Y − 9 = X 3 1 ​ ⋅ Y − 9 1 ​ = X 3 Y − 9 1 ​ What Is Tyler's Error?A. Both Powers Should Be In The Numerator With Positive

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Understanding the Concept of Negative Exponents

In mathematics, negative exponents are a crucial concept that can be easily misunderstood. A negative exponent is a shorthand way of expressing a fraction with a positive exponent in the denominator. For instance, x−3x^{-3} is equivalent to 1x3\frac{1}{x^3}. However, when simplifying expressions involving negative exponents, it's essential to follow the correct procedures to avoid errors.

Tyler's Procedure

Tyler's procedure for simplifying the expression x−3y−9x^{-3} y^{-9} is as follows:

x−3y−9=1x3⋅1y−9=1x3y−9x^{-3} y^{-9} = \frac{1}{x^3} \cdot \frac{1}{y^{-9}} = \frac{1}{x^3 y^{-9}}

Identifying the Error

At first glance, Tyler's procedure may seem correct. However, upon closer inspection, we can identify the error. The mistake lies in the way Tyler handled the negative exponent y−9y^{-9}. When simplifying the expression, Tyler incorrectly placed the negative exponent in the denominator, resulting in 1x3y−9\frac{1}{x^3 y^{-9}}.

Correcting the Error

To correct Tyler's error, we need to apply the correct rules for simplifying expressions with negative exponents. When a negative exponent is present, we can move it to the numerator by changing its sign. In this case, we can rewrite y−9y^{-9} as 1y9\frac{1}{y^9}.

Correct Procedure

The correct procedure for simplifying the expression x−3y−9x^{-3} y^{-9} is as follows:

x−3y−9=1x3⋅1y9=1x3y9x^{-3} y^{-9} = \frac{1}{x^3} \cdot \frac{1}{y^9} = \frac{1}{x^3 y^9}

Key Takeaways

  1. Understanding Negative Exponents: Negative exponents are a shorthand way of expressing fractions with positive exponents in the denominator.
  2. Correct Procedure: When simplifying expressions with negative exponents, move the negative exponent to the numerator by changing its sign.
  3. Error Identification: Tyler's error was placing the negative exponent in the denominator, resulting in an incorrect simplification.

Conclusion

In conclusion, Tyler's error in simplifying the expression x−3y−9x^{-3} y^{-9} was due to an incorrect handling of the negative exponent. By applying the correct rules for simplifying expressions with negative exponents, we can arrive at the correct solution. This article has provided a step-by-step guide on how to simplify expressions with negative exponents and has highlighted the importance of understanding this concept in mathematics.

Additional Examples

To further reinforce the concept of negative exponents, let's consider a few additional examples:

  • x−2y3=1x2⋅y3=y3x2x^{-2} y^3 = \frac{1}{x^2} \cdot y^3 = \frac{y^3}{x^2}
  • x2y−4=x2y4x^2 y^{-4} = \frac{x^2}{y^4}
  • x−5y−2=1x5⋅1y2=1x5y2x^{-5} y^{-2} = \frac{1}{x^5} \cdot \frac{1}{y^2} = \frac{1}{x^5 y^2}

Practice Problems

To practice simplifying expressions with negative exponents, try the following problems:

  • Simplify the expression x−4y2x^{-4} y^2.
  • Simplify the expression x3y−6x^3 y^{-6}.
  • Simplify the expression x−2y−3x^{-2} y^{-3}.

Answer Key

  • x−4y2=1x4⋅y2=y2x4x^{-4} y^2 = \frac{1}{x^4} \cdot y^2 = \frac{y^2}{x^4}
  • x3y−6=x3y6x^3 y^{-6} = \frac{x^3}{y^6}
  • x−2y−3=1x2⋅1y3=1x2y3x^{-2} y^{-3} = \frac{1}{x^2} \cdot \frac{1}{y^3} = \frac{1}{x^2 y^3}
    Frequently Asked Questions (FAQs) on Simplifying Expressions with Negative Exponents =====================================================================================

Q: What is a negative exponent?

A: A negative exponent is a shorthand way of expressing a fraction with a positive exponent in the denominator. For instance, x−3x^{-3} is equivalent to 1x3\frac{1}{x^3}.

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

A: To simplify an expression with a negative exponent, move the negative exponent to the numerator by changing its sign. For example, x−3y−9=1x3⋅1y9=1x3y9x^{-3} y^{-9} = \frac{1}{x^3} \cdot \frac{1}{y^9} = \frac{1}{x^3 y^9}.

Q: What is the rule for simplifying expressions with negative exponents?

A: The rule for simplifying expressions with negative exponents is to move the negative exponent to the numerator by changing its sign. This can be represented as:

x−n=1xnx^{-n} = \frac{1}{x^n}

Q: Can you provide an example of simplifying an expression with a negative exponent?

A: Let's consider the expression x−2y3x^{-2} y^3. To simplify this expression, we can move the negative exponent to the numerator by changing its sign:

x−2y3=1x2⋅y3=y3x2x^{-2} y^3 = \frac{1}{x^2} \cdot y^3 = \frac{y^3}{x^2}

Q: How do I handle multiple negative exponents in an expression?

A: When handling multiple negative exponents in an expression, we can simplify each negative exponent separately and then multiply the results. For example, let's consider the expression x−3y−9z2x^{-3} y^{-9} z^2. We can simplify this expression as follows:

x−3y−9z2=1x3⋅1y9⋅z2=z2x3y9x^{-3} y^{-9} z^2 = \frac{1}{x^3} \cdot \frac{1}{y^9} \cdot z^2 = \frac{z^2}{x^3 y^9}

Q: What is the difference between a negative exponent and a fraction?

A: A negative exponent is a shorthand way of expressing a fraction with a positive exponent in the denominator. For instance, x−3x^{-3} is equivalent to 1x3\frac{1}{x^3}. However, a fraction is a separate mathematical concept that represents a part of a whole.

Q: Can you provide a real-world example of using negative exponents?

A: Negative exponents have numerous applications in real-world scenarios, particularly in physics and engineering. For instance, the formula for the force of gravity between two objects is given by:

F=G⋅m1⋅m2r2F = \frac{G \cdot m_1 \cdot m_2}{r^2}

where FF is the force of gravity, GG is the gravitational constant, m1m_1 and m2m_2 are the masses of the objects, and rr is the distance between them. In this formula, the negative exponent r2r^2 represents the inverse square law of gravity.

Q: How do I practice simplifying expressions with negative exponents?

A: To practice simplifying expressions with negative exponents, try the following exercises:

  • Simplify the expression x−4y2x^{-4} y^2.
  • Simplify the expression x3y−6x^3 y^{-6}.
  • Simplify the expression x−2y−3x^{-2} y^{-3}.

Answer Key

  • x−4y2=1x4⋅y2=y2x4x^{-4} y^2 = \frac{1}{x^4} \cdot y^2 = \frac{y^2}{x^4}
  • x3y−6=x3y6x^3 y^{-6} = \frac{x^3}{y^6}
  • x−2y−3=1x2⋅1y3=1x2y3x^{-2} y^{-3} = \frac{1}{x^2} \cdot \frac{1}{y^3} = \frac{1}{x^2 y^3}