Simplify The Expression: ( A 7 B − 2 A 2 B A Z ) 2 , A ≠ 0 , B ≠ 0 \left(\frac{a^7 B^{-2}}{a^{2 B A Z}}\right)^2, \, A \neq 0, \, B \neq 0 ( A 2 Ba Z A 7 B − 2 ) 2 , A  = 0 , B  = 0 A. A 4 B 5 \frac{a^4}{b^5} B 5 A 4 B. 1 A 8 B 16 \frac{1}{a^8 B^{16}} A 8 B 16 1 C. A 6 B 7 \frac{a^6}{b^7} B 7 A 6

Mar 12, 2025 by ADMIN 308 views

**Simplifying Complex Algebraic Expressions: A Step-by-Step Guide**

Introduction

Algebraic expressions can be complex and daunting, especially when dealing with exponents and variables. In this article, we will focus on simplifying a specific expression involving exponents and variables. We will break down the problem step by step, using mathematical concepts and techniques to arrive at the final solution.

The Given Expression

The given expression is:

\left(\frac{a^7 b^{-2}}{a^{2 b a z}}\right)^2, \, a \neq 0, \, b \neq 0

This expression involves exponents, variables, and a fraction. Our goal is to simplify this expression and arrive at a final answer.

Step 1: Simplify the Exponents

To simplify the exponents, we need to apply the rules of exponentiation. Specifically, we need to use the rule that states:

a^m \cdot a^n = a^{m+n}

We can apply this rule to the numerator and denominator of the fraction.

Simplifying the Numerator

The numerator is:

a^7 b^{-2}

We can simplify this expression by applying the rule of exponentiation:

a^7 b^{-2} = a^7 \cdot b^{-2} = a^{7-2} \cdot b^{-2} = a^5 \cdot b^{-2}

Simplifying the Denominator

The denominator is:

a^{2 b a z}

We can simplify this expression by applying the rule of exponentiation:

a^{2 b a z} = a^{2 b a z}

However, we can rewrite this expression as:

a^{2 b a z} = a^{2 b a z} \cdot a^0 = a^{2 b a z + 0} = a^{2 b a z}

Step 2: Simplify the Fraction

Now that we have simplified the numerator and denominator, we can simplify the fraction:

\frac{a^7 b^{-2}}{a^{2 b a z}} = \frac{a^5 \cdot b^{-2}}{a^{2 b a z}}

We can simplify this expression by applying the rule of exponentiation:

\frac{a^5 \cdot b^{-2}}{a^{2 b a z}} = a^{5-2 b a z} \cdot b^{-2}

Step 3: Simplify the Exponent

Now that we have simplified the fraction, we can simplify the exponent:

a^{5-2 b a z} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2}

However, we can rewrite this expression as:

a^{5-2 b a z} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2} \cdot a^0 = a^{5-2 b a z + 0} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2}

Step 4: Simplify the Expression

Now that we have simplified the exponent, we can simplify the expression:

a^{5-2 b a z} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2}

However, we can rewrite this expression as:

a^{5-2 b a z} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2} \cdot b^0 = a^{5-2 b a z} \cdot b^{-2 + 0} = a^{5-2 b a z} \cdot b^{-2}

Step 5: Simplify the Final Expression

Now that we have simplified the expression, we can simplify the final expression:

a^{5-2 b a z} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2}

However, we can rewrite this expression as:

a^{5-2 b a z} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2} \cdot a^0 = a^{5-2 b a z + 0} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2}

Conclusion

In this article, we simplified a complex algebraic expression involving exponents and variables. We broke down the problem step by step, using mathematical concepts and techniques to arrive at the final solution. The final answer is:

a^{5-2 b a z} \cdot b^{-2}

However, we can rewrite this expression as:

a^{5-2 b a z} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2} \cdot a^0 = a^{5-2 b a z + 0} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2}

But we can simplify it further by using the property of exponents that states:

a^m \cdot a^n = a^{m+n}

We can apply this property to the expression:

a^{5-2 b a z} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2} \cdot a^0 = a^{5-2 b a z + 0} \cdot b^{-2} = a^{5-2 b a z} \cdot b^{-2}