$(4 x^4 \cdot x^4)^3$
Answer (2)
Multiply the terms inside the parenthesis: 4 x 4 ⋅ x 4 = 4 x 8 .
Apply the power of a product rule: ( 4 x 8 ) 3 = 4 3 ( x 8 ) 3 .
Calculate 4 3 = 64 and simplify the exponent: ( x 8 ) 3 = x 24 .
The simplified expression is 64 x 24 .
Explanation
Initial Analysis Let's simplify the given expression step by step. We will use the properties of exponents to simplify the expression inside the parenthesis first, and then apply the power of a product rule.
Simplifying Inside Parenthesis First, we simplify the expression inside the parenthesis: 4 x 4 ⋅ x 4 = 4 ⋅ ( x 4 ⋅ x 4 ) When multiplying terms with the same base, we add the exponents: x 4 ⋅ x 4 = x 4 + 4 = x 8 So, the expression inside the parenthesis becomes: 4 x 8
Applying Power of a Product Rule Now, we raise the entire expression to the power of 3: ( 4 x 8 ) 3 We apply the power of a product rule, which states that ( ab ) n = a n b n :
( 4 x 8 ) 3 = 4 3 ⋅ ( x 8 ) 3
Simplifying the Powers We know that 4 3 = 4 ⋅ 4 ⋅ 4 = 64 . Now we simplify ( x 8 ) 3 . When raising a power to a power, we multiply the exponents: ( x 8 ) 3 = x 8 ⋅ 3 = x 24
Final Simplification Putting it all together, we have: 4 3 ⋅ ( x 8 ) 3 = 64 x 24 So, the simplified expression is 64 x 24 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the volume of a cube with side length 2 x 2 , the volume would be ( 2 x 2 ) 3 = 8 x 6 . This type of calculation is essential in determining material requirements or understanding scaling effects in various systems. Similarly, in computer science, understanding exponents helps in analyzing the complexity of algorithms, where the runtime might be expressed as O ( n 2 ) or O ( 2 n ) , indicating how the execution time scales with the input size.
To simplify ( 4 x 4 ⋅ x 4 ) 3 , first combine the terms inside the parenthesis to get 4 x 8 . Then apply the power of a product rule to find that it equals 64 x 24 . Thus, the final simplified expression is (64x^{24}.
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