Simplify:$\[ X^{\frac{1}{2}} Y^2 \cdot X^{\frac{3}{2}} Y^{\frac{1}{2}} \\]A. \[$ 4 X^{\frac{5}{2}} Y^{\frac{13}{4}} \$\] B. \[$ Y^4 X^{\frac{18}{6}} \$\] C. \[$ X Y^{\frac{7}{2}} \$\] D. \[$ X^2 Y^{\frac{5}{2}}

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, focusing on the given expression: x12y2â‹…x32y12x^{\frac{1}{2}} y^2 \cdot x^{\frac{3}{2}} y^{\frac{1}{2}}. We will break down the steps involved in simplifying this expression and provide a clear understanding of the underlying concepts.

Understanding Exponents

Before we dive into simplifying the given expression, it's essential to understand the concept of exponents. Exponents are a shorthand way of representing repeated multiplication. For example, x2x^2 can be read as "x squared" or "x multiplied by itself twice". In the given expression, we have exponents with fractional values, such as 12\frac{1}{2} and 32\frac{3}{2}. These exponents represent the power to which the base number is raised.

Simplifying the Expression

To simplify the given expression, we need to apply the rules of exponents. The expression can be rewritten as:

x12y2â‹…x32y12=x12+32â‹…y2+12x^{\frac{1}{2}} y^2 \cdot x^{\frac{3}{2}} y^{\frac{1}{2}} = x^{\frac{1}{2} + \frac{3}{2}} \cdot y^{2 + \frac{1}{2}}

Using the rule of exponents that states amâ‹…an=am+na^m \cdot a^n = a^{m+n}, we can simplify the expression further:

x12+32â‹…y2+12=x2â‹…y52x^{\frac{1}{2} + \frac{3}{2}} \cdot y^{2 + \frac{1}{2}} = x^2 \cdot y^{\frac{5}{2}}

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options provided:

  • Option A: 4x52y1344 x^{\frac{5}{2}} y^{\frac{13}{4}}
  • Option B: y4x186y^4 x^{\frac{18}{6}}
  • Option C: xy72x y^{\frac{7}{2}}
  • Option D: x2y52x^2 y^{\frac{5}{2}}

Comparing the simplified expression with the options, we can see that:

  • Option A is incorrect because it has an extra factor of 4 and a different exponent for y.
  • Option B is incorrect because it has a different exponent for x and an extra factor of y.
  • Option C is incorrect because it has a different exponent for x and an extra factor of y.
  • Option D is correct because it matches the simplified expression exactly.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By understanding the concept of exponents and applying the rules of exponents, we can simplify complex expressions and arrive at the correct solution. In this article, we simplified the expression x12y2â‹…x32y12x^{\frac{1}{2}} y^2 \cdot x^{\frac{3}{2}} y^{\frac{1}{2}} and evaluated the options provided. We found that the correct solution is Option D: x2y52x^2 y^{\frac{5}{2}}.

Final Answer

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, focusing on the given expression: x12y2â‹…x32y12x^{\frac{1}{2}} y^2 \cdot x^{\frac{3}{2}} y^{\frac{1}{2}}. We broke down the steps involved in simplifying this expression and provided a clear understanding of the underlying concepts. In this article, we will address some common questions and concerns related to simplifying algebraic expressions.

Q&A

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

A: When simplifying expressions with the same base, we can add or subtract the exponents. For example, x2â‹…x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5.

Q: How do I simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, we can rewrite the expression using the rule amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}. For example, x12=xx^{\frac{1}{2}} = \sqrt{x}.

Q: What is the difference between a product and a quotient of expressions?

A: A product of expressions is the result of multiplying two or more expressions together, while a quotient of expressions is the result of dividing one expression by another. For example, x2â‹…y3x^2 \cdot y^3 is a product, while x2y3\frac{x^2}{y^3} is a quotient.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we can rewrite the expression using the rule a−m=1ama^{-m} = \frac{1}{a^m}. For example, x−2=1x2x^{-2} = \frac{1}{x^2}.

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

A: When simplifying expressions with exponents and coefficients, we can multiply the coefficients and add the exponents. For example, 2x2â‹…3x3=6x2+3=6x52x^2 \cdot 3x^3 = 6x^{2+3} = 6x^5.

Q: How do I simplify expressions with radicals?

A: To simplify expressions with radicals, we can rewrite the expression using the rule amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}. For example, x2=x22=x\sqrt{x^2} = x^{\frac{2}{2}} = x.

Q: What is the difference between a simplified expression and a factored expression?

A: A simplified expression is an expression that has been reduced to its simplest form, while a factored expression is an expression that has been written as a product of simpler expressions. For example, x2+2x+1x^2 + 2x + 1 is a simplified expression, while (x+1)2(x+1)^2 is a factored expression.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By understanding the rules and concepts involved in simplifying expressions, we can arrive at the correct solution and gain a deeper understanding of the underlying mathematics. In this article, we addressed some common questions and concerns related to simplifying algebraic expressions, providing a clear and concise guide for anyone looking to improve their math skills.

Final Tips

  • Always start by simplifying the expression using the rules of exponents.
  • Use the rule amâ‹…an=am+na^m \cdot a^n = a^{m+n} to simplify expressions with the same base.
  • Rewrite expressions with fractional exponents using the rule amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}.
  • Use the rule a−m=1ama^{-m} = \frac{1}{a^m} to simplify expressions with negative exponents.
  • Multiply coefficients and add exponents when simplifying expressions with exponents and coefficients.
  • Rewrite expressions with radicals using the rule amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.

By following these tips and practicing regularly, you can become proficient in simplifying algebraic expressions and tackle even the most complex math problems with confidence.