Classwork: Dividing ExponentsSimplify The Expression:$\[ \frac{(3xy)^2 \times (x^4y^2)^3}{7^0 \times 3(y^2)^2} \\]

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Understanding Exponents and Their Rules

In mathematics, exponents are a shorthand way of representing repeated multiplication of a number. For example, x3{x^3} means x×x×x{x \times x \times x}. Exponents are a crucial concept in algebra and are used extensively in various mathematical operations. In this article, we will focus on dividing exponents and simplifying expressions that involve exponents.

The Rules of Exponents

Before we dive into dividing exponents, it's essential to understand the basic rules of exponents. These rules will help us simplify expressions and perform operations involving exponents.

  • Product of Powers Rule: When multiplying two powers with the same base, we add the exponents. For example, x2×x3=x2+3=x5{x^2 \times x^3 = x^{2+3} = x^5}.
  • Power of a Power Rule: When raising a power to another power, we multiply the exponents. For example, (x2)3=x2×3=x6{(x^2)^3 = x^{2 \times 3} = x^6}.
  • Quotient of Powers Rule: When dividing two powers with the same base, we subtract the exponents. For example, x2x3=x2−3=x−1{\frac{x^2}{x^3} = x^{2-3} = x^{-1}}.
  • Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, x0=1{x^0 = 1}.

Dividing Exponents

Now that we have a solid understanding of the rules of exponents, let's focus on dividing exponents. When dividing two powers with the same base, we subtract the exponents. For example, x2x3=x2−3=x−1{\frac{x^2}{x^3} = x^{2-3} = x^{-1}}.

Simplifying the Expression

Let's simplify the given expression:

(3xy)2×(x4y2)370×3(y2)2{ \frac{(3xy)^2 \times (x^4y^2)^3}{7^0 \times 3(y^2)^2} }

To simplify this expression, we need to apply the rules of exponents. We will start by simplifying the numerator and denominator separately.

Simplifying the Numerator

The numerator is (3xy)2×(x4y2)3{(3xy)^2 \times (x^4y^2)^3}. We can simplify this expression by applying the product of powers rule and the power of a power rule.

(3xy)2×(x4y2)3=(3xy)2×(x4)3×(y2)3{ (3xy)^2 \times (x^4y^2)^3 = (3xy)^2 \times (x^4)^3 \times (y^2)^3 }

=32×x2×y2×x4×3×y2×3{ = 3^2 \times x^2 \times y^2 \times x^{4 \times 3} \times y^{2 \times 3} }

=9×x2×y2×x12×y6{ = 9 \times x^2 \times y^2 \times x^{12} \times y^6 }

=9×x2+12×y2+6{ = 9 \times x^{2+12} \times y^{2+6} }

=9×x14×y8{ = 9 \times x^{14} \times y^8 }

Simplifying the Denominator

The denominator is 70×3(y2)2{7^0 \times 3(y^2)^2}. We can simplify this expression by applying the zero exponent rule and the power of a power rule.

70×3(y2)2=1×3×y2×2{ 7^0 \times 3(y^2)^2 = 1 \times 3 \times y^{2 \times 2} }

=3×y4{ = 3 \times y^4 }

Simplifying the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator.

(3xy)2×(x4y2)370×3(y2)2=9×x14×y83×y4{ \frac{(3xy)^2 \times (x^4y^2)^3}{7^0 \times 3(y^2)^2} = \frac{9 \times x^{14} \times y^8}{3 \times y^4} }

=93×x141×y8y4{ = \frac{9}{3} \times \frac{x^{14}}{1} \times \frac{y^8}{y^4} }

=3×x14×y8−4{ = 3 \times x^{14} \times y^{8-4} }

=3×x14×y4{ = 3 \times x^{14} \times y^4 }

Therefore, the simplified expression is 3×x14×y4{3 \times x^{14} \times y^4}.

Conclusion

In this article, we have learned about dividing exponents and simplifying expressions that involve exponents. We have applied the rules of exponents, including the product of powers rule, the power of a power rule, the quotient of powers rule, and the zero exponent rule. We have also simplified a given expression by applying these rules. By understanding and applying the rules of exponents, we can simplify complex expressions and perform mathematical operations with ease.

Practice Problems

  1. Simplify the expression: (2x3y2)2×(x4y3)350×2(y3)2{\frac{(2x^3y^2)^2 \times (x^4y^3)^3}{5^0 \times 2(y^3)^2}}
  2. Simplify the expression: (3x2y4)2×(x3y2)370×3(y4)2{\frac{(3x^2y^4)^2 \times (x^3y^2)^3}{7^0 \times 3(y^4)^2}}
  3. Simplify the expression: (2x3y2)3×(x4y3)250×2(y3)3{\frac{(2x^3y^2)^3 \times (x^4y^3)^2}{5^0 \times 2(y^3)^3}}

Answer Key

  1. 4×x8×y8{4 \times x^8 \times y^8}
  2. 9×x10×y10{9 \times x^{10} \times y^10}
  3. 4×x12×y6{4 \times x^{12} \times y^6}
    Classwork: Dividing Exponents Q&A =====================================

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about dividing exponents and simplifying expressions that involve exponents.

Q: What is the rule for dividing exponents?

A: When dividing two powers with the same base, we subtract the exponents. For example, x2x3=x2−3=x−1{\frac{x^2}{x^3} = x^{2-3} = x^{-1}}.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the rules of exponents, including the product of powers rule, the power of a power rule, the quotient of powers rule, and the zero exponent rule.

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, we add the exponents. For example, x2×x3=x2+3=x5{x^2 \times x^3 = x^{2+3} = x^5}.

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, we multiply the exponents. For example, (x2)3=x2×3=x6{(x^2)^3 = x^{2 \times 3} = x^6}.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two powers with the same base, we subtract the exponents. For example, x2x3=x2−3=x−1{\frac{x^2}{x^3} = x^{2-3} = x^{-1}}.

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=1{x^0 = 1}.

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

A: To simplify an expression with multiple exponents, you need to apply the rules of exponents in the correct order. For example, (3xy)2×(x4y2)370×3(y2)2{\frac{(3xy)^2 \times (x^4y^2)^3}{7^0 \times 3(y^2)^2}} can be simplified by first simplifying the numerator and denominator separately, and then simplifying the resulting expression.

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid when simplifying expressions with exponents include:

  • Not applying the rules of exponents in the correct order
  • Not simplifying the numerator and denominator separately
  • Not canceling out common factors
  • Not checking for errors in the final answer

Q: How can I practice simplifying expressions with exponents?

A: You can practice simplifying expressions with exponents by working through practice problems, such as the ones listed below:

  1. Simplify the expression: (2x3y2)2×(x4y3)350×2(y3)2{\frac{(2x^3y^2)^2 \times (x^4y^3)^3}{5^0 \times 2(y^3)^2}}
  2. Simplify the expression: (3x2y4)2×(x3y2)370×3(y4)2{\frac{(3x^2y^4)^2 \times (x^3y^2)^3}{7^0 \times 3(y^4)^2}}
  3. Simplify the expression: (2x3y2)3×(x4y3)250×2(y3)3{\frac{(2x^3y^2)^3 \times (x^4y^3)^2}{5^0 \times 2(y^3)^3}}

Practice Problems

  1. Simplify the expression: (2x3y2)2×(x4y3)350×2(y3)2{\frac{(2x^3y^2)^2 \times (x^4y^3)^3}{5^0 \times 2(y^3)^2}}
  2. Simplify the expression: (3x2y4)2×(x3y2)370×3(y4)2{\frac{(3x^2y^4)^2 \times (x^3y^2)^3}{7^0 \times 3(y^4)^2}}
  3. Simplify the expression: (2x3y2)3×(x4y3)250×2(y3)3{\frac{(2x^3y^2)^3 \times (x^4y^3)^2}{5^0 \times 2(y^3)^3}}

Answer Key

  1. 4×x8×y8{4 \times x^8 \times y^8}
  2. 9×x10×y10{9 \times x^{10} \times y^{10}}
  3. 4×x12×y6{4 \times x^{12} \times y^6}

Conclusion

In this article, we have addressed some of the most frequently asked questions about dividing exponents and simplifying expressions that involve exponents. We have also provided practice problems and an answer key to help you practice simplifying expressions with exponents. By understanding and applying the rules of exponents, you can simplify complex expressions and perform mathematical operations with ease.