Simplify The Expression: ((-8 X⁸ Y⁴ Z⁵)/(20 X⁵ Y⁴ Z³))⁴

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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. In this article, we will delve into the world of algebraic manipulation and explore the steps involved in simplifying the expression: ((-8 x⁸ y⁴ z⁵)/(20 x⁵ y⁴ z³))⁴. We will break down the expression into manageable parts, apply various algebraic rules, and ultimately arrive at a simplified form.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its components. The expression consists of a fraction, where the numerator is (-8 x⁸ y⁴ z⁵) and the denominator is (20 x⁵ y⁴ z³). The entire expression is raised to the power of 4.

((-8 x⁸ y⁴ z⁵)/(20 x⁵ y⁴ z³))⁴

Step 1: Simplify the Numerator and Denominator

To simplify the expression, we can start by simplifying the numerator and denominator separately. We can begin by factoring out common terms from both the numerator and denominator.

(-8 x⁸ y⁴ z⁵) = -8 x⁵ x³ y⁴ z⁵
(20 x⁵ y⁴ z³) = 20 x⁵ y⁴ z³

Step 2: Cancel Out Common Terms

Now that we have factored out common terms, we can cancel them out to simplify the expression. The common terms are x⁵, y⁴, and z³.

(-8 x⁵ x³ y⁴ z⁵) / (20 x⁵ y⁴ z³) = (-8 x³ z²) / (20)

Step 3: Raise the Simplified Expression to the Power of 4

Now that we have simplified the expression, we can raise it to the power of 4.

((-8 x³ z²) / (20))⁴ = ((-8 x³ z²)⁴) / (20⁴)

Step 4: Simplify the Exponents

To simplify the exponents, we can apply the power rule of exponents, which states that (aⁿ)ᵐ = a^(n*m).

((-8 x³ z²)⁴) = (-8)⁴ (x³)⁴ (z²)⁴
= 4096 x¹² z⁸
(20⁴) = 160000

Step 5: Simplify the Expression

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

(4096 x¹² z⁸) / (160000) = 4096/160000 x¹² z⁸
= 256/10000 x¹² z⁸
= 0.0256 x¹² z⁸

Conclusion

In this article, we have simplified the expression ((-8 x⁸ y⁴ z⁵)/(20 x⁵ y⁴ z³))⁴ by applying various algebraic rules and techniques. We broken down the expression into manageable parts, factored out common terms, canceled them out, raised the simplified expression to the power of 4, simplified the exponents, and finally arrived at a simplified form. The simplified expression is 0.0256 x¹² z⁸.

Final Answer

The final answer is: 0.0256 x¹² z⁸

Frequently Asked Questions

  • Q: What is the simplified form of the expression ((-8 x⁸ y⁴ z⁵)/(20 x⁵ y⁴ z³))⁴? A: The simplified form of the expression is 0.0256 x¹² z⁸.
  • Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, you can apply various algebraic rules and techniques, such as factoring out common terms, canceling them out, and raising the simplified expression to the power of 4.
  • Q: What is the power rule of exponents? A: The power rule of exponents states that (aⁿ)ᵐ = a^(n*m).

Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying structure. In this article, we will delve into the world of algebraic manipulation and explore the steps involved in simplifying algebraic expressions. We will also answer some frequently asked questions to help you better understand the process.

Q&A: Algebraic Expression Simplification

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

A: The first step in simplifying an algebraic expression is to identify and factor out common terms. This can help you simplify the expression by canceling out common factors.

Q: How do I identify common terms in an algebraic expression?

A: To identify common terms in an algebraic expression, look for terms that have the same variable or constant factor. For example, in the expression 2x + 3x, the common term is x.

Q: What is the difference between a variable and a constant in an algebraic expression?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change. For example, in the expression 2x + 3, x is a variable and 3 is a constant.

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, you can use the same techniques as before, such as factoring out common terms and canceling them out. However, you may need to use additional techniques, such as combining like terms or using the distributive property.

Q: What is the distributive property in algebra?

A: The distributive property is a rule that allows you to multiply a single term by multiple terms. For example, in the expression 2(x + 3), you can use the distributive property to multiply 2 by each term inside the parentheses: 2x + 6.

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

A: To simplify an algebraic expression with exponents, you can use the power rule of exponents, which states that (aⁿ)ᵐ = a^(n*m). For example, in the expression (2x)², you can use the power rule to simplify the expression: 4x².

Q: What is the power rule of exponents?

A: The power rule of exponents states that (aⁿ)ᵐ = a^(n*m). This means that when you raise a power to another power, you can multiply the exponents.

Q: How do I simplify an algebraic expression with fractions?

A: To simplify an algebraic expression with fractions, you can use the same techniques as before, such as factoring out common terms and canceling them out. However, you may need to use additional techniques, such as multiplying the numerator and denominator by the same value to eliminate the fraction.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be simplified to a fraction, while an irrational expression is an expression that cannot be simplified to a fraction.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can use the same techniques as before, such as factoring out common terms and canceling them out. However, you may need to use additional techniques, such as multiplying the numerator and denominator by the same value to eliminate the fraction.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to check your work and make sure that the expression is in its simplest form.

Conclusion

In this article, we have explored the steps involved in simplifying algebraic expressions and answered some frequently asked questions. We have discussed the importance of identifying and factoring out common terms, using the distributive property, and applying the power rule of exponents. We have also covered the differences between rational and irrational expressions and provided tips for simplifying rational expressions. By following these steps and techniques, you can simplify even the most complex algebraic expressions.

Final Answer

The final answer is: 0.0256 x¹² z⁸

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