%5E3(ab%5E3)%5E2%2F(a%5E2b%5E2)%5E4.jpeg "(3a^2b)^3(ab^3)^2/(a^2b^2)^4")
Simplifying Algebraic Expressions: (3a^2b)^3(ab^3)^2/(a^2b^2)^4
This article will guide you through simplifying the algebraic expression (3a^2b)^3(ab^3)^2/(a^2b^2)^4. We'll break down the process step by step to make it easy to understand.
Understanding the Rules
Before we begin, let's review some key rules of exponents:
- Product of powers: x<sup>m</sup> * x<sup>n</sup> = x<sup>m+n</sup>
- Power of a product: (xy)<sup>m</sup> = x<sup>m</sup> * y<sup>m</sup>
- Power of a power: (x<sup>m</sup>)<sup>n</sup> = x<sup>m*n</sup>
- Quotient of powers: x<sup>m</sup> / x<sup>n</sup> = x<sup>m-n</sup>
Simplifying the Expression
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Expand the powers:
(3a<sup>2</sup>b)<sup>3</sup> = 3<sup>3</sup> * a<sup>2*3</sup> * b<sup>3</sup> = 27a<sup>6</sup>b<sup>3</sup>
(ab<sup>3</sup>)<sup>2</sup> = a<sup>2</sup> * b<sup>3*2</sup> = a<sup>2</sup>b<sup>6</sup>
(a<sup>2</sup>b<sup>2</sup>)<sup>4</sup> = a<sup>24</sup> * b<sup>24</sup> = a<sup>8</sup>b<sup>8</sup>
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Substitute the expanded terms back into the original expression:
(27a<sup>6</sup>b<sup>3</sup>)(a<sup>2</sup>b<sup>6</sup>) / (a<sup>8</sup>b<sup>8</sup>)
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Combine like terms in the numerator:
27a<sup>6+2</sup>b<sup>3+6</sup> / (a<sup>8</sup>b<sup>8</sup>) = 27a<sup>8</sup>b<sup>9</sup> / (a<sup>8</sup>b<sup>8</sup>)
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Apply the quotient of powers rule:
27a<sup>8-8</sup>b<sup>9-8</sup> = 27a<sup>0</sup>b<sup>1</sup>
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Simplify further, knowing a<sup>0</sup> = 1:
27 * 1 * b = 27b
Conclusion
Therefore, the simplified form of the expression (3a<sup>2</sup>b)<sup>3</sup>(ab<sup>3</sup>)<sup>2</sup>/(a<sup>2</sup>b<sup>2</sup>)<sup>4</sup> is 27b. By understanding the rules of exponents and applying them systematically, we can effectively simplify complex algebraic expressions.