Which of these is equivalent to $9^2+4^2$?
A. $(9+9)+(4+4)$
B. $(9 \times 9)+(4 \times 4)$
C. $(9+4)^2$
D. $(9 \times 4)^2$
Answer (2)
Calculate 9 2 + 4 2 = 81 + 16 = 97 .
Evaluate the options: ( 9 + 9 ) + ( 4 + 4 ) = 26 , ( 9 × 9 ) + ( 4 × 4 ) = 97 , ( 9 + 4 ) 2 = 169 , ( 9 × 4 ) 2 = 1296 .
Compare the results and find the equivalent expression.
The equivalent expression is ( 9 × 9 ) + ( 4 × 4 ) .
Explanation
Problem Analysis We are given the expression 9 2 + 4 2 and four options. Our goal is to find which of the options is equivalent to the given expression.
Calculating the Expression First, let's calculate the value of the expression 9 2 + 4 2 . We have: 9 2 + 4 2 = 9 × 9 + 4 × 4 = 81 + 16 = 97
Evaluating the Options Now, let's evaluate each of the given options:
Option 1: ( 9 + 9 ) + ( 4 + 4 ) = 18 + 8 = 26 Option 2: ( 9 × 9 ) + ( 4 × 4 ) = 81 + 16 = 97 Option 3: ( 9 + 4 ) 2 = ( 13 ) 2 = 169 Option 4: ( 9 × 4 ) 2 = ( 36 ) 2 = 1296
Finding the Equivalent Expression Comparing the value of 9 2 + 4 2 with the values of the options, we see that:
9 2 + 4 2 = 97 Option 1: 26 = 97 Option 2: 97 = 97 Option 3: 169 = 97 Option 4: 1296 = 97
Therefore, the expression equivalent to 9 2 + 4 2 is ( 9 × 9 ) + ( 4 × 4 ) .
Conclusion The expression equivalent to 9 2 + 4 2 is ( 9 × 9 ) + ( 4 × 4 ) .
Examples
Understanding numerical equivalence is crucial in many fields, such as engineering and physics, where calculations need to be simplified or rearranged without changing their values. For instance, when calculating the total energy in a system, you might encounter an expression like 9 2 + 4 2 . Recognizing that this is equivalent to ( 9 × 9 ) + ( 4 × 4 ) allows for easier computation or manipulation of the equation, ensuring accurate results in complex simulations or designs.
The equivalent expression for 9 2 + 4 2 is Option B: ( 9 × 9 ) + ( 4 × 4 ) , which evaluates to 97. The calculations show that the other options do not equal 97. Therefore, Option B is the correct choice.
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