(b) Without using a calculator, evaluate $54^2-44^2$.

Recognize the expression as a difference of squares: a 2 − b 2 .
Apply the difference of squares factorization: 5 4 2 − 4 4 2 = ( 54 + 44 ) ( 54 − 44 ) .
Simplify the expression: ( 54 + 44 ) ( 54 − 44 ) = ( 98 ) ( 10 ) .
Multiply to obtain the final answer: ( 98 ) ( 10 ) = 980 ​ .

Explanation

Recognizing the Pattern We are asked to evaluate 5 4 2 − 4 4 2 without using a calculator. This looks like a perfect opportunity to use the difference of squares factorization!

Applying the Difference of Squares Recall that the difference of squares factorization is given by a 2 − b 2 = ( a + b ) ( a − b ) . We can apply this to our expression with a = 54 and b = 44 .

Setting up the Calculation So, we have 5 4 2 − 4 4 2 = ( 54 + 44 ) ( 54 − 44 ) . Now we just need to perform the addition and subtraction.

Performing Addition First, let's calculate 54 + 44 . We can break this down as 50 + 40 + 4 + 4 = 90 + 8 = 98 .

Performing Subtraction Next, let's calculate 54 − 44 . This is simply 10 .

Final Calculation Now we multiply the results: ( 98 ) ( 10 ) = 980 .

The Answer Therefore, 5 4 2 − 4 4 2 = 980 .


Examples
The difference of squares is a useful concept in many areas, such as engineering and physics. For example, when calculating the difference in kinetic energy between two objects with slightly different velocities, the difference of squares factorization can simplify the calculation. Imagine you're designing a rollercoaster. Using the difference of squares can help you quickly calculate changes in energy as the cars move, ensuring a thrilling but safe ride. This method simplifies complex calculations, making design adjustments easier and more efficient.

Answered by GinnyAnswer | 2025-07-03

To evaluate 5 4 2 − 4 4 2 , we use the difference of squares formula to get ( 54 + 44 ) ( 54 − 44 ) = 98 × 10 = 980 . Thus, the final answer is 980 . This method simplifies our calculation significantly.
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Answered by Anonymous | 2025-07-04