Select The Best Answer For The Question.1. Simplify $\frac{x^0 Y^{-3}}{x^2 Y^{-1}}$A. $\frac{y^3}{2}$B. \$\frac{1}{x^2 Y^2}$[/tex\]C. $\frac{4 Y^3}{x^3}$D. $\frac{y}{x^6}$

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Understanding Exponents and Simplification

Exponents are a fundamental concept in mathematics, and simplifying expressions with exponents is a crucial skill to master. In this article, we will delve into the world of exponents and provide a step-by-step guide on how to simplify the given expression: $\frac{x^0 y{-3}}{x2 y^{-1}}$

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ means $x \times x \times x$, and $x^4$ means $x \times x \times x \times x$. Exponents can be positive or negative, and they can also be fractional.

Simplifying Exponents: A Step-by-Step Guide

To simplify the given expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: There are no parentheses in the given expression, so we can move on to the next step.
  2. Exponents: We need to simplify the exponents in the numerator and denominator.
  3. Multiplication and Division: We need to simplify the multiplication and division operations.

Step 1: Simplify the Exponents

Let's start by simplifying the exponents in the numerator and denominator.

x0y−3x2y−1\frac{x^0 y^{-3}}{x^2 y^{-1}}

We can simplify the exponents as follows:

  • x^0$ means $1$, so we can replace $x^0$ with $1$.

  • y^{-3}$ means $\frac{1}{y^3}$, so we can replace $y^{-3}$ with $\frac{1}{y^3}$.

  • x^2$ remains the same.

  • y^{-1}$ means $\frac{1}{y}$, so we can replace $y^{-1}$ with $\frac{1}{y}$.

The expression now becomes:

1×1y3x2×1y\frac{1 \times \frac{1}{y^3}}{x^2 \times \frac{1}{y}}

Step 2: Simplify the Multiplication and Division

Now that we have simplified the exponents, we can simplify the multiplication and division operations.

1×1y3x2×1y\frac{1 \times \frac{1}{y^3}}{x^2 \times \frac{1}{y}}

We can simplify the multiplication and division operations as follows:

  • 1 \times \frac{1}{y^3}$ means $\frac{1}{y^3}$.

  • x^2 \times \frac{1}{y}$ means $\frac{x^2}{y}$.

The expression now becomes:

1y3x2y\frac{\frac{1}{y^3}}{\frac{x^2}{y}}

Step 3: Simplify the Division

Now that we have simplified the multiplication and division operations, we can simplify the division operation.

1y3x2y\frac{\frac{1}{y^3}}{\frac{x^2}{y}}

We can simplify the division operation as follows:

  • \frac{\frac{1}{y^3}}{\frac{x^2}{y}}$ means $\frac{1}{y^3} \times \frac{y}{x^2}$.

  • \frac{1}{y^3} \times \frac{y}{x^2}$ means $\frac{1}{x^2 y^2}$.

The final simplified expression is:

1x2y2\frac{1}{x^2 y^2}

Conclusion

In this article, we have provided a step-by-step guide on how to simplify the given expression: $\frac{x^0 y{-3}}{x2 y^{-1}}$. We have used the order of operations (PEMDAS) to simplify the exponents, multiplication, and division operations. The final simplified expression is $\frac{1}{x^2 y^2}$.

Answer

The correct answer is:

B. $\frac{1}{x^2 y^2}$

Discussion

This problem requires a good understanding of exponents and simplification. The key concept is to simplify the exponents and then simplify the multiplication and division operations. The order of operations (PEMDAS) is crucial in simplifying the expression.

Practice Problems

Here are some practice problems to help you reinforce your understanding of exponents and simplification:

  1. Simplify $\frac{x^2 y3}{x4 y^2}$
  2. Simplify $\frac{x^{-2} y4}{x3 y^{-1}}$
  3. Simplify $\frac{x^0 y2}{x2 y^0}$

References

Q: What are exponents?

A: Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ means $x \times x \times x$, and $x^4$ means $x \times x \times x \times x$.

Q: How do I simplify exponents?

A: To simplify exponents, you need to follow the order of operations (PEMDAS):

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

Q: What is the order of operations (PEMDAS)?

A: PEMDAS is a mnemonic device that helps you remember the order of operations:

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

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, you can use the following rule:

x−n=1xnx^{-n} = \frac{1}{x^n}

For example, $x^{-2} = \frac{1}{x^2}$.

Q: How do I simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, you can use the following rule:

xmn=xmnx^{\frac{m}{n}} = \sqrt[n]{x^m}

For example, $x^{\frac{1}{2}} = \sqrt{x}$.

Q: What is the difference between $x^2$ and $x{23}$?

A: $x^2$ means $x \times x$, while $x{23}$ means $x \times x \times x \times x \times x \times x$.

Q: How do I simplify expressions with multiple exponents?

A: To simplify expressions with multiple exponents, you need to follow the order of operations (PEMDAS) and simplify the exponents from left to right.

For example, $x^2 \times x^3 = x^{2+3} = x^5$.

Q: Can I simplify expressions with variables in the exponent?

A: Yes, you can simplify expressions with variables in the exponent. For example, $x^{2y} = (x2)y$.

Q: How do I simplify expressions with exponents and fractions?

A: To simplify expressions with exponents and fractions, you need to follow the order of operations (PEMDAS) and simplify the exponents and fractions from left to right.

For example, $\frac{x2}{x3} = x^{2-3} = x^{-1} = \frac{1}{x}$.

Conclusion

In this article, we have answered some frequently asked questions on simplifying exponents. We have covered topics such as the order of operations (PEMDAS), simplifying negative exponents, fractional exponents, and multiple exponents. We hope this article has been helpful in clarifying any doubts you may have had on simplifying exponents.