Which expression is equal to the expression below? $(7 \cdot 2)^7$
A. $7 \cdot(7 \cdot 2)$
B. $7^7 \cdot 2^7$
C. $9^7$
D. $7^7+2^7$
Answer (2)
Apply the power of a product rule: ( a c d o t b ) n = a n c d o t b n .
Apply the rule to the given expression: ( 7 c d o t 2 ) 7 = 7 7 c d o t 2 7 .
Compare the result with the given options.
The expression equal to ( 7 c d o t 2 ) 7 is 7 7 c d o t 2 7 .
Explanation
Understanding the Problem We are given the expression ( 7 c d o t 2 ) 7 and asked to find an equivalent expression from the options provided.
Applying the Power of a Product Rule To solve this, we need to use the power of a product rule, which states that for any numbers a and b , and any exponent n , the following is true: ( a c d o t b ) n = a n c d o t b n
Simplifying the Expression Applying this rule to our expression, we get: ( 7 c d o t 2 ) 7 = 7 7 c d o t 2 7
Comparing with the Options Now we compare our simplified expression 7 7 c d o t 2 7 to the given options:
A. 7 c d o t ( 7 c d o t 2 ) is not equal to 7 7 c d o t 2 7 .
B. 7 7 c d o t 2 7 is equal to 7 7 c d o t 2 7 .
C. 9 7 is not equal to 7 7 c d o t 2 7 .
D. 7 7 + 2 7 is not equal to 7 7 c d o t 2 7 .
Final Answer Therefore, the expression equal to ( 7 c d o t 2 ) 7 is 7 7 c d o t 2 7 .
Examples
The power of a product rule is useful in various real-life scenarios. For instance, if you are calculating the area of multiple square gardens that have sides which are multiples of a certain unit, you can use this rule to simplify the calculations. Suppose you have 5 square gardens, each with sides of length 2 x . The total area would be 5 c d o t ( 2 x ) 2 = 5 c d o t ( 4 x 2 ) = 20 x 2 . This simplifies the calculation and makes it easier to understand the relationship between the side length and the total area.
The expression ( 7 ⋅ 2 ) 7 simplifies to 7 7 ⋅ 2 7 using the power of a product rule. Comparing it with the options, the equivalent expression is option B: 7 7 ⋅ 2 7 . Therefore, the answer is B.
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