What is the simplest form of $\sqrt[3]{27 a^3 b^7}$?
A. $3 a b^2(\sqrt[3]{b})$
B. $3 a b^3(\sqrt[3]{3 a b})$
C. $9 a b^2(\sqrt[3]{b})$
D. $9 a b^3(\sqrt[3]{3 a b})$
Answer (2)
Rewrite the expression using fractional exponents: ( 27 a 3 b 7 ) 3 1 .
Express 27 as 3 3 and b 7 as b 6 ⋅ b .
Apply the power of a product rule and simplify the exponents.
Rewrite the final simplified expression: 3 a b 2 3 b .
Explanation
Understanding the Problem We are asked to simplify the expression 3 27 a 3 b 7 . This involves finding the cube root of the given expression, which includes constants and variables raised to certain powers. We will use properties of exponents and radicals to simplify it.
Rewriting with Fractional Exponents First, rewrite the expression using fractional exponents: 3 27 a 3 b 7 = ( 27 a 3 b 7 ) 3 1 .
Expressing as Perfect Cubes Next, express 27 as 3 3 and rewrite b 7 as b 6 + 1 = b 6 ⋅ b = ( b 2 ) 3 ⋅ b . This allows us to identify perfect cubes within the expression.
Combining Terms Now, rewrite the entire expression: ( 27 a 3 b 7 ) 3 1 = ( 3 3 a 3 b 6 b ) 3 1 .
Applying the Power of a Product Rule Apply the power of a product rule, which states that ( x y ) n = x n y n . This gives us ( 3 3 a 3 b 6 b ) 3 1 = ( 3 3 ) 3 1 ( a 3 ) 3 1 ( b 6 ) 3 1 ( b ) 3 1 .
Simplifying Exponents Simplify the exponents: ( 3 3 ) 3 1 ( a 3 ) 3 1 ( b 6 ) 3 1 ( b ) 3 1 = 3 3 ⋅ 3 1 a 3 ⋅ 3 1 b 6 ⋅ 3 1 b 3 1 = 3 1 a 1 b 2 b 3 1 = 3 a b 2 b 3 1 .
Final Simplified Expression Finally, rewrite the expression using radical notation: 3 a b 2 b 3 1 = 3 a b 2 3 b .
Conclusion Therefore, the simplest form of 3 27 a 3 b 7 is 3 a b 2 3 b .
Examples
Imagine you are designing a storage container, and its volume is given by 3 27 a 3 b 7 . Simplifying this expression to 3 a b 2 3 b helps you understand the key dimensions that influence the container's size. This is useful in optimizing the design for efficient space utilization. For example, if a represents the height and b relates to the width, knowing the simplified form allows you to quickly calculate the volume for different values of a and b , aiding in practical applications such as packaging or warehouse design.
The simplest form of 3 27 a 3 b 7 is 3 a b 2 3 b . This means the correct answer is option A.
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