Simplify $\frac{x^0 Y^{-3}}{x^2 Y^{-1}}$.A. $\frac{y^3}{2}$ B. $\frac{1}{x^2 Y^2}$ C. $\frac{4 Y^3}{x^3}$ D. $\frac{y}{x^6}$

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Understanding Exponents and Their Rules

Exponents are a fundamental concept in algebra, and understanding how to simplify expressions with exponents is crucial for solving mathematical problems. In this article, we will focus on simplifying the expression x0y−3x2y−1\frac{x^0 y^{-3}}{x^2 y^{-1}} using the rules of exponents.

The Rules of Exponents

Before we dive into simplifying the given expression, let's review the basic rules of exponents:

  • Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}.
  • Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, (xa)b=xab(x^a)^b = x^{ab}.
  • Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. For example, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.
  • Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, x0=1x^0 = 1.
  • Negative Exponent Rule: 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}.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the expression x0y−3x2y−1\frac{x^0 y^{-3}}{x^2 y^{-1}}.

First, we can simplify the numerator using the zero exponent rule. Since x0=1x^0 = 1, the numerator becomes 1⋅y−3=y−31 \cdot y^{-3} = y^{-3}.

Next, we can simplify the denominator using the quotient of powers rule. Since we are dividing two powers with the same base, we subtract the exponents. Therefore, the denominator becomes x2−0⋅y−1−(−3)=x2⋅y2x^{2-0} \cdot y^{-1-(-3)} = x^2 \cdot y^2.

Now, we can rewrite the expression as y−3x2⋅y2\frac{y^{-3}}{x^2 \cdot y^2}. To simplify further, we can use the negative exponent rule to rewrite the numerator as 1y3\frac{1}{y^3}.

Therefore, the simplified expression is 1y3÷x2⋅y2=1y3⋅1x2⋅y2=1x2⋅y5\frac{1}{y^3} \div x^2 \cdot y^2 = \frac{1}{y^3} \cdot \frac{1}{x^2 \cdot y^2} = \frac{1}{x^2 \cdot y^5}.

Conclusion

In conclusion, simplifying the expression x0y−3x2y−1\frac{x^0 y^{-3}}{x^2 y^{-1}} requires applying the rules of exponents. By using the product of powers rule, power of a power rule, quotient of powers rule, zero exponent rule, and negative exponent rule, we can simplify the expression to 1x2⋅y5\frac{1}{x^2 \cdot y^5}.

Answer

Understanding Exponents and Their Rules

Exponents are a fundamental concept in algebra, and understanding how to simplify expressions with exponents is crucial for solving mathematical problems. In this article, we will focus on simplifying the expression x0y−3x2y−1\frac{x^0 y^{-3}}{x^2 y^{-1}} using the rules of exponents.

Q&A: Simplifying Exponents

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, x0=1x^0 = 1.

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

A: To simplify an expression with a negative exponent, you can use the negative exponent rule. This rule states that 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 product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, add the exponents. For example, xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}.

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

A: To simplify an expression with multiple exponents, you can use the quotient of powers rule. This rule states that when dividing two powers with the same base, subtract the exponents. For example, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.

Q: What is the power of a power rule?

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

Q: How do I simplify an expression with a zero exponent in the numerator?

A: To simplify an expression with a zero exponent in the numerator, you can use the zero exponent rule. This rule states that any non-zero number raised to the power of zero is equal to 1. For example, x0=1x^0 = 1.

Q: What is the correct answer for the expression x0y−3x2y−1\frac{x^0 y^{-3}}{x^2 y^{-1}}?

A: The correct answer is 1x2⋅y5\boxed{\frac{1}{x^2 \cdot y^5}}.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not applying the zero exponent rule: Remember that any non-zero number raised to the power of zero is equal to 1.
  • Not using the negative exponent rule: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base.
  • Not applying the product of powers rule: When multiplying two powers with the same base, add the exponents.
  • Not applying the quotient of powers rule: When dividing two powers with the same base, subtract the exponents.
  • Not applying the power of a power rule: When raising a power to another power, multiply the exponents.

Conclusion

Simplifying expressions with exponents requires a solid understanding of the rules of exponents. By applying the product of powers rule, power of a power rule, quotient of powers rule, zero exponent rule, and negative exponent rule, you can simplify even the most complex expressions. Remember to avoid common mistakes and always double-check your work.

Practice Problems

Try these practice problems to test your skills:

  1. Simplify the expression x2y3x4y2\frac{x^2 y^3}{x^4 y^2}.
  2. Simplify the expression x−3y2x2y−1\frac{x^{-3} y^2}{x^2 y^{-1}}.
  3. Simplify the expression (x2y3)4x6y6\frac{(x^2 y^3)^4}{x^6 y^6}.

Answer Key

  1. yx2\boxed{\frac{y}{x^2}}
  2. y3x5\boxed{\frac{y^3}{x^5}}
  3. y6x2\boxed{\frac{y^6}{x^2}}