Simplify The Expression:${ \frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}} \div \frac{2 \sqrt[3]{-13824}}{\sqrt[3]{5832}} }$

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Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying structure and meaning. In this article, we will delve into the world of algebraic manipulation and explore the steps involved in simplifying a given expression. We will use the expression 31382432−156253÷2−13824358323\frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}} \div \frac{2 \sqrt[3]{-13824}}{\sqrt[3]{5832}} as a case study and demonstrate how to simplify it using various algebraic techniques.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its components. The expression consists of two fractions, each containing cube roots. The first fraction is 31382432−156253\frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}}, while the second fraction is 2−13824358323\frac{2 \sqrt[3]{-13824}}{\sqrt[3]{5832}}. To simplify the expression, we need to manipulate these fractions using various algebraic techniques.

Simplifying the Cube Roots

The first step in simplifying the expression is to simplify the cube roots. We can start by factoring the numbers inside the cube roots. For example, 13824=28⋅3413824 = 2^8 \cdot 3^4, while −15625=−56-15625 = -5^6. Similarly, −13824=−28⋅34-13824 = -2^8 \cdot 3^4, and 5832=24⋅365832 = 2^4 \cdot 3^6. By factoring these numbers, we can rewrite the cube roots as:

138243=28⋅343=283⋅343\sqrt[3]{13824} = \sqrt[3]{2^8 \cdot 3^4} = 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}

−156253=−563=−52\sqrt[3]{-15625} = \sqrt[3]{-5^6} = -5^2

−138243=−28⋅343=−283⋅343\sqrt[3]{-13824} = \sqrt[3]{-2^8 \cdot 3^4} = -2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}

58323=24⋅363=243⋅32\sqrt[3]{5832} = \sqrt[3]{2^4 \cdot 3^6} = 2^{\frac{4}{3}} \cdot 3^2

Simplifying the Fractions

Now that we have simplified the cube roots, we can simplify the fractions. We can start by simplifying the first fraction:

31382432−156253=3⋅283⋅3432⋅−52\frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}} = \frac{3 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}{2 \cdot -5^2}

=3⋅283⋅343−2⋅52= \frac{3 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}{-2 \cdot 5^2}

=−3⋅283⋅3432⋅52= -\frac{3 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}{2 \cdot 5^2}

Similarly, we can simplify the second fraction:

2−13824358323=2⋅−283⋅343243⋅32\frac{2 \sqrt[3]{-13824}}{\sqrt[3]{5832}} = \frac{2 \cdot -2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}{2^{\frac{4}{3}} \cdot 3^2}

=2⋅−283⋅343243⋅32= \frac{2 \cdot -2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}{2^{\frac{4}{3}} \cdot 3^2}

=−243⋅343= -2^{\frac{4}{3}} \cdot 3^{\frac{4}{3}}

Dividing the Fractions

Now that we have simplified the fractions, we can divide them. To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction. In this case, we can multiply the first fraction by the reciprocal of the second fraction:

31382432−156253÷2−13824358323=31382432−156253⋅583232−138243\frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}} \div \frac{2 \sqrt[3]{-13824}}{\sqrt[3]{5832}} = \frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}} \cdot \frac{\sqrt[3]{5832}}{2 \sqrt[3]{-13824}}

=3⋅283⋅343−2⋅52⋅243⋅32−283⋅343= \frac{3 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}{-2 \cdot 5^2} \cdot \frac{2^{\frac{4}{3}} \cdot 3^2}{-2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

Canceling Out Terms

Now that we have multiplied the fractions, we can cancel out terms. We can start by canceling out the common terms in the numerator and denominator:

=3⋅283⋅343−2⋅52⋅243⋅32−283⋅343= \frac{3 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}{-2 \cdot 5^2} \cdot \frac{2^{\frac{4}{3}} \cdot 3^2}{-2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅283⋅343⋅243⋅32−2⋅52⋅−283⋅343= \frac{3 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}} \cdot 2^{\frac{4}{3}} \cdot 3^2}{-2 \cdot 5^2 \cdot -2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅2123⋅363⋅32−2⋅52⋅−283⋅343= \frac{3 \cdot 2^{\frac{12}{3}} \cdot 3^{\frac{6}{3}} \cdot 3^2}{-2 \cdot 5^2 \cdot -2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅24⋅33−2⋅52⋅−283⋅343= \frac{3 \cdot 2^4 \cdot 3^3}{-2 \cdot 5^2 \cdot -2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

Simplifying the Expression

Now that we have canceled out terms, we can simplify the expression further. We can start by simplifying the numerator:

=3⋅24⋅33−2⋅52⋅−283⋅343= \frac{3 \cdot 2^4 \cdot 3^3}{-2 \cdot 5^2 \cdot -2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅−283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot -2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

=3⋅16⋅27−2⋅25⋅283⋅343= \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}

= \frac{3 \cdot 16 \cdot 27<br/> # Simplify the Expression: A Comprehensive Guide to Algebraic Manipulation - Q&A

Introduction

In our previous article, we explored the steps involved in simplifying a given expression. We used the expression 31382432−156253÷2−13824358323\frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}} \div \frac{2 \sqrt[3]{-13824}}{\sqrt[3]{5832}} as a case study and demonstrated how to simplify it using various algebraic techniques. In this article, we will answer some of the most frequently asked questions related to simplifying expressions.

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

A: The first step in simplifying an expression is to identify the components of the expression. This includes identifying the numbers, variables, and mathematical operations involved in the expression.

Q: How do I simplify cube roots?

A: To simplify cube roots, you can factor the numbers inside the cube roots. For example, 13824=28⋅3413824 = 2^8 \cdot 3^4, while −15625=−56-15625 = -5^6. By factoring these numbers, you can rewrite the cube roots as 138243=28⋅343=283⋅343\sqrt[3]{13824} = \sqrt[3]{2^8 \cdot 3^4} = 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}} and −156253=−563=−52\sqrt[3]{-15625} = \sqrt[3]{-5^6} = -5^2.

Q: How do I simplify fractions?

A: To simplify fractions, you can start by simplifying the numerator and denominator separately. For example, in the expression 31382432−156253\frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}}, you can simplify the numerator as 3⋅283⋅3433 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}} and the denominator as −2⋅52-2 \cdot 5^2.

Q: How do I divide fractions?

A: To divide fractions, you can multiply the first fraction by the reciprocal of the second fraction. In this case, you can multiply the first fraction by the reciprocal of the second fraction: 31382432−156253÷2−13824358323=31382432−156253⋅583232−138243\frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}} \div \frac{2 \sqrt[3]{-13824}}{\sqrt[3]{5832}} = \frac{3 \sqrt[3]{13824}}{2 \sqrt[3]{-15625}} \cdot \frac{\sqrt[3]{5832}}{2 \sqrt[3]{-13824}}.

Q: How do I cancel out terms?

A: To cancel out terms, you can start by identifying the common terms in the numerator and denominator. For example, in the expression 3⋅283⋅343−2⋅52⋅243⋅32−283⋅343\frac{3 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}{-2 \cdot 5^2} \cdot \frac{2^{\frac{4}{3}} \cdot 3^2}{-2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}, you can cancel out the common terms 2832^{\frac{8}{3}} and 3433^{\frac{4}{3}}.

Q: What is the final simplified expression?

A: The final simplified expression is 3⋅16⋅27−2⋅25⋅283⋅343\frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}}. However, we can simplify this expression further by canceling out the common terms 2832^{\frac{8}{3}} and 3433^{\frac{4}{3}}.

Q: How do I simplify the final expression?

A: To simplify the final expression, you can start by canceling out the common terms 2832^{\frac{8}{3}} and 3433^{\frac{4}{3}}. This will give you the final simplified expression: $\frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac{8}{3}} \cdot 3^{\frac{4}{3}}} = \frac{3 \cdot 16 \cdot 27}{-2 \cdot 25 \cdot 2^{\frac