Simplify The Expression: 2 ( 3 Y X 2 ) 3 × 81 Y 2 24 Y 8 3 2\left(3 Y X^2\right)^3 \times \sqrt[3]{\frac{81 Y^2}{24 Y^8}} 2 ( 3 Y X 2 ) 3 × 3 24 Y 8 81 Y 2 ​ ​

by ADMIN 160 views

Introduction

Mathematical expressions can be complex and daunting, but with the right approach, they can be simplified to reveal their underlying structure. In this article, we will focus on simplifying the given expression: 2(3yx2)3×81y224y832\left(3 y x^2\right)^3 \times \sqrt[3]{\frac{81 y^2}{24 y^8}}. We will break down the expression into manageable parts, apply the necessary mathematical operations, and finally arrive at the simplified form.

Understanding the Expression

The given expression involves several mathematical operations, including exponentiation, multiplication, and radicals. To simplify the expression, we need to understand the properties of exponents and radicals, as well as the rules for multiplying and dividing expressions.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. In the given expression, we have the term (3yx2)3\left(3 y x^2\right)^3, which involves raising the expression 3yx23 y x^2 to the power of 3.

Radicals

Radicals are mathematical expressions that involve the square root or cube root of a number. In the given expression, we have the term 81y224y83\sqrt[3]{\frac{81 y^2}{24 y^8}}, which involves taking the cube root of the fraction 81y224y8\frac{81 y^2}{24 y^8}.

Simplifying the Expression

To simplify the expression, we will start by evaluating the exponentiation and radical terms separately.

Evaluating the Exponentiation Term

The exponentiation term is (3yx2)3\left(3 y x^2\right)^3. To evaluate this term, we need to apply the power rule of exponents, which states that (ab)n=anbn(ab)^n = a^nb^n. Applying this rule, we get:

(3yx2)3=33y3x6=27y3x6\left(3 y x^2\right)^3 = 3^3 y^3 x^6 = 27 y^3 x^6

Evaluating the Radical Term

The radical term is 81y224y83\sqrt[3]{\frac{81 y^2}{24 y^8}}. To evaluate this term, we need to apply the rule for radicals, which states that an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}. Applying this rule, we get:

81y224y83=(81y224y8)13=8113y232413y83=3y23243y83\sqrt[3]{\frac{81 y^2}{24 y^8}} = \left(\frac{81 y^2}{24 y^8}\right)^{\frac{1}{3}} = \frac{81^{\frac{1}{3}} y^{\frac{2}{3}}}{24^{\frac{1}{3}} y^{\frac{8}{3}}} = \frac{3 y^{\frac{2}{3}}}{2^{\frac{4}{3}} y^{\frac{8}{3}}}

Simplifying the Expression

Now that we have evaluated the exponentiation and radical terms, we can simplify the expression by multiplying the two terms together.

2(3yx2)3×81y224y83=2×27y3x6×3y23243y832\left(3 y x^2\right)^3 \times \sqrt[3]{\frac{81 y^2}{24 y^8}} = 2 \times 27 y^3 x^6 \times \frac{3 y^{\frac{2}{3}}}{2^{\frac{4}{3}} y^{\frac{8}{3}}}

To simplify this expression, we need to apply the rules for multiplying and dividing expressions. We can start by canceling out any common factors between the numerator and denominator.

2×27y3x6×3y23243y83=2×27×3×y3×x6×y23×1243×y832 \times 27 y^3 x^6 \times \frac{3 y^{\frac{2}{3}}}{2^{\frac{4}{3}} y^{\frac{8}{3}}} = 2 \times 27 \times 3 \times y^3 \times x^6 \times y^{\frac{2}{3}} \times \frac{1}{2^{\frac{4}{3}} \times y^{\frac{8}{3}}}

Now, we can simplify the expression by combining like terms.

2×27×3×y3×x6×y23×1243×y83=162y3×x6×y23×1243×y832 \times 27 \times 3 \times y^3 \times x^6 \times y^{\frac{2}{3}} \times \frac{1}{2^{\frac{4}{3}} \times y^{\frac{8}{3}}} = 162 y^3 \times x^6 \times y^{\frac{2}{3}} \times \frac{1}{2^{\frac{4}{3}} \times y^{\frac{8}{3}}}

To simplify this expression further, we need to apply the rule for multiplying exponents with the same base. This rule states that am×an=am+na^m \times a^n = a^{m+n}. Applying this rule, we get:

162y3×x6×y23×1243×y83=162×x6×y3+23×1243×y83162 y^3 \times x^6 \times y^{\frac{2}{3}} \times \frac{1}{2^{\frac{4}{3}} \times y^{\frac{8}{3}}} = 162 \times x^6 \times y^{3+\frac{2}{3}} \times \frac{1}{2^{\frac{4}{3}} \times y^{\frac{8}{3}}}

Now, we can simplify the expression by combining like terms.

162×x6×y3+23×1243×y83=162×x6×y113×1243×y83162 \times x^6 \times y^{3+\frac{2}{3}} \times \frac{1}{2^{\frac{4}{3}} \times y^{\frac{8}{3}}} = 162 \times x^6 \times y^{\frac{11}{3}} \times \frac{1}{2^{\frac{4}{3}} \times y^{\frac{8}{3}}}

To simplify this expression further, we need to apply the rule for dividing exponents with the same base. This rule states that aman=am−n\frac{a^m}{a^n} = a^{m-n}. Applying this rule, we get:

162×x6×y113×1243×y83=162×x6×y113−83×1243162 \times x^6 \times y^{\frac{11}{3}} \times \frac{1}{2^{\frac{4}{3}} \times y^{\frac{8}{3}}} = 162 \times x^6 \times y^{\frac{11}{3}-\frac{8}{3}} \times \frac{1}{2^{\frac{4}{3}}}

Now, we can simplify the expression by combining like terms.

162×x6×y113−83×1243=162×x6×y33×1243162 \times x^6 \times y^{\frac{11}{3}-\frac{8}{3}} \times \frac{1}{2^{\frac{4}{3}}} = 162 \times x^6 \times y^{\frac{3}{3}} \times \frac{1}{2^{\frac{4}{3}}}

To simplify this expression further, we need to apply the rule for dividing exponents with the same base. This rule states that aman=am−n\frac{a^m}{a^n} = a^{m-n}. Applying this rule, we get:

162×x6×y33×1243=162×x6×y1×1243162 \times x^6 \times y^{\frac{3}{3}} \times \frac{1}{2^{\frac{4}{3}}} = 162 \times x^6 \times y^1 \times \frac{1}{2^{\frac{4}{3}}}

Now, we can simplify the expression by combining like terms.

162×x6×y1×1243=162×x6×y×1243162 \times x^6 \times y^1 \times \frac{1}{2^{\frac{4}{3}}} = 162 \times x^6 \times y \times \frac{1}{2^{\frac{4}{3}}}

To simplify this expression further, we need to apply the rule for multiplying fractions. This rule states that ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}. Applying this rule, we get:

162×x6×y×1243=162×x6×y243162 \times x^6 \times y \times \frac{1}{2^{\frac{4}{3}}} = \frac{162 \times x^6 \times y}{2^{\frac{4}{3}}}

Now, we can simplify the expression by evaluating the numerator and denominator separately.

162×x6×y243=162×x6×y2×223\frac{162 \times x^6 \times y}{2^{\frac{4}{3}}} = \frac{162 \times x^6 \times y}{2 \times 2^{\frac{2}{3}}}

To simplify this expression further, we need to apply the rule for dividing fractions. This rule states that ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}. Applying this rule, we get:

162×x6×y2×223=162×x6×y2×1223\frac{162 \times x^6 \times y}{2 \times 2^{\frac{2}{3}}} = \frac{162 \times x^6 \times y}{2} \times \frac{1}{2^{\frac{2}{3}}}

Now, we can simplify the expression by combining like terms.

162×x6×y2×1223=81×x6×y×1223\frac{162 \times x^6 \times y}{2} \times \frac{1}{2^{\frac{2}{3}}} = 81 \times x^6 \times y \times \frac{1}{2^{\frac{2}{3}}}

To simplify this expression further, we need to apply the rule for multiplying fractions. This rule states that $\frac{a}{b} \times

Introduction

In our previous article, we walked through the process of simplifying the given mathematical expression: 2(3yx2)3×81y224y832\left(3 y x^2\right)^3 \times \sqrt[3]{\frac{81 y^2}{24 y^8}}. We broke down the expression into manageable parts, applied the necessary mathematical operations, and finally arrived at the simplified form. In this article, we will answer some of the most frequently asked questions related to the simplification of the given expression.

Q&A

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to evaluate the exponentiation term, (3yx2)3\left(3 y x^2\right)^3. This involves applying the power rule of exponents, which states that (ab)n=anbn(ab)^n = a^nb^n.

Q: How do I apply the power rule of exponents?

A: To apply the power rule of exponents, you need to raise each factor in the expression to the power of the exponent. In this case, you would raise 33, yy, and x2x^2 to the power of 33.

Q: What is the next step in simplifying the given expression?

A: The next step in simplifying the given expression is to evaluate the radical term, 81y224y83\sqrt[3]{\frac{81 y^2}{24 y^8}}. This involves applying the rule for radicals, which states that an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}.

Q: How do I apply the rule for radicals?

A: To apply the rule for radicals, you need to raise the expression inside the radical to the power of the reciprocal of the index. In this case, you would raise 81y224y8\frac{81 y^2}{24 y^8} to the power of 13\frac{1}{3}.

Q: What is the final step in simplifying the given expression?

A: The final step in simplifying the given expression is to multiply the two terms together, 2(3yx2)32\left(3 y x^2\right)^3 and 81y224y83\sqrt[3]{\frac{81 y^2}{24 y^8}}. This involves applying the rules for multiplying and dividing expressions.

Q: How do I apply the rules for multiplying and dividing expressions?

A: To apply the rules for multiplying and dividing expressions, you need to follow the order of operations (PEMDAS). This means that you need to evaluate the expressions inside the parentheses first, then evaluate any exponents, and finally multiply or divide the expressions.

Q: What is the simplified form of the given expression?

A: The simplified form of the given expression is 162×x6×y243\frac{162 \times x^6 \times y}{2^{\frac{4}{3}}}. This involves applying the rules for multiplying and dividing fractions.

Q: How do I evaluate the numerator and denominator of the fraction?

A: To evaluate the numerator and denominator of the fraction, you need to follow the order of operations (PEMDAS). This means that you need to evaluate any exponents first, then multiply or divide the expressions.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 81×x6×y223\frac{81 \times x^6 \times y}{2^{\frac{2}{3}}}. This involves applying the rules for multiplying and dividing fractions.

Conclusion

Simplifying mathematical expressions can be a challenging task, but with the right approach, it can be done. In this article, we walked through the process of simplifying the given expression: 2(3yx2)3×81y224y832\left(3 y x^2\right)^3 \times \sqrt[3]{\frac{81 y^2}{24 y^8}}. We answered some of the most frequently asked questions related to the simplification of the given expression and provided a step-by-step guide to evaluating the expression.

Frequently Asked Questions

  • Q: What is the first step in simplifying the given expression? A: The first step in simplifying the given expression is to evaluate the exponentiation term, (3yx2)3\left(3 y x^2\right)^3.
  • Q: How do I apply the power rule of exponents? A: To apply the power rule of exponents, you need to raise each factor in the expression to the power of the exponent.
  • Q: What is the next step in simplifying the given expression? A: The next step in simplifying the given expression is to evaluate the radical term, 81y224y83\sqrt[3]{\frac{81 y^2}{24 y^8}}.
  • Q: How do I apply the rule for radicals? A: To apply the rule for radicals, you need to raise the expression inside the radical to the power of the reciprocal of the index.
  • Q: What is the final step in simplifying the given expression? A: The final step in simplifying the given expression is to multiply the two terms together, 2(3yx2)32\left(3 y x^2\right)^3 and 81y224y83\sqrt[3]{\frac{81 y^2}{24 y^8}}.
  • Q: How do I apply the rules for multiplying and dividing expressions? A: To apply the rules for multiplying and dividing expressions, you need to follow the order of operations (PEMDAS).
  • Q: What is the simplified form of the given expression? A: The simplified form of the given expression is 162×x6×y243\frac{162 \times x^6 \times y}{2^{\frac{4}{3}}}.
  • Q: How do I evaluate the numerator and denominator of the fraction? A: To evaluate the numerator and denominator of the fraction, you need to follow the order of operations (PEMDAS).
  • Q: What is the final answer to the given expression? A: The final answer to the given expression is 81×x6×y223\frac{81 \times x^6 \times y}{2^{\frac{2}{3}}}.

Additional Resources

  • For more information on simplifying mathematical expressions, please refer to our previous article on the topic.
  • For more information on the power rule of exponents, please refer to our article on the topic.
  • For more information on the rule for radicals, please refer to our article on the topic.
  • For more information on the rules for multiplying and dividing expressions, please refer to our article on the topic.