%5E2(a%5E2b)%5E3.jpeg "(ab^3)^2(a^2b)^3")
Simplifying Algebraic Expressions: (ab^3)^2(a^2b)^3
This article will walk you through the process of simplifying the algebraic expression (ab^3)^2(a^2b)^3. We'll break down each step, using the rules of exponents to arrive at the simplest form.
Understanding the Rules of Exponents
Before we begin, let's review the key rules of exponents that we'll be using:
- Product of powers: x<sup>m</sup> * x<sup>n</sup> = x<sup>m+n</sup>
- Power of a product: (xy)<sup>n</sup> = x<sup>n</sup> * y<sup>n</sup>
- Power of a power: (x<sup>m</sup>)<sup>n</sup> = x<sup>m*n</sup>
Step-by-Step Simplification
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Apply the power of a product rule to both expressions:
(ab<sup>3</sup>)<sup>2</sup> = a<sup>2</sup>b<sup>6</sup> (a<sup>2</sup>b)<sup>3</sup> = a<sup>6</sup>b<sup>3</sup>
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Substitute the simplified terms back into the original expression:
(a<sup>2</sup>b<sup>6</sup>)(a<sup>6</sup>b<sup>3</sup>)
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Apply the product of powers rule for terms with the same base:
a<sup>2+6</sup>b<sup>6+3</sup>
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Simplify the exponents:
a<sup>8</sup>b<sup>9</sup>
Final Result
Therefore, the simplified form of (ab<sup>3</sup>)<sup>2</sup>(a<sup>2</sup>b)<sup>3</sup> is a<sup>8</sup>b<sup>9</sup>.