Math Problem Statement

Simplify \(\frac{(x-a)^{-2}}{(x-a)^{-4}} \cdot (x-a)^0 = (x-a)?\).

Solution

Let's solve the given expression step by step:

Problem:

(xa)2(xa)4(xa)0=(xa)?\frac{(x-a)^{-2}}{(x-a)^{-4}} \cdot (x-a)^0 = (x-a)?

Step 1: Simplify the fraction

Using the property of exponents aman=amn\frac{a^m}{a^n} = a^{m-n}: (xa)2(xa)4=(xa)2(4)=(xa)2+4=(xa)2.\frac{(x-a)^{-2}}{(x-a)^{-4}} = (x-a)^{-2 - (-4)} = (x-a)^{-2 + 4} = (x-a)^2.

Step 2: Multiply with (xa)0(x-a)^0

The property of exponents states that a0=1a^0 = 1 (for a0a \neq 0). Hence: (xa)2(xa)0=(xa)21=(xa)2.(x-a)^2 \cdot (x-a)^0 = (x-a)^2 \cdot 1 = (x-a)^2.

Step 3: Check the right-hand side

The given expression simplifies to: (xa)2=(xa)?(x-a)^2 = (x-a)?

Conclusion:

The exponent on the right-hand side must be 22. Thus: (xa)2=(xa)2.(x-a)^2 = (x-a)^2.

Final Answer:

The given equation is true.

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Algebraic Simplification

Formulas

\(\frac{a^m}{a^n} = a^{m-n}\)
\(a^0 = 1\) (for \(a \neq 0\))

Theorems

Properties of Exponents

Suitable Grade Level

Grades 9-10