Sure, let's break down each part of your question step-by-step for clear understanding.
### Part a: Simplify [tex]\( 25^2 - 15^2 \)[/tex]
This expression fits the form of a difference of squares, which is given by the formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, [tex]\( a = 25 \)[/tex] and [tex]\( b = 15 \)[/tex]. Applying the formula:
[tex]\[ 25^2 - 15^2 = (25 - 15)(25 + 15) \][/tex]
Now, let's compute the values inside the parentheses:
[tex]\[ 25 - 15 = 10 \][/tex]
[tex]\[ 25 + 15 = 40 \][/tex]
Therefore:
[tex]\[ 25^2 - 15^2 = 10 \times 40 \][/tex]
Multiplying these together gives:
[tex]\[ 10 \times 40 = 400 \][/tex]
So, the simplified result of [tex]\( 25^2 - 15^2 \)[/tex] is:
[tex]\[ 400 \][/tex]
### Part e: Simplify [tex]\( 51 \times 49 \)[/tex]
Notice that [tex]\( 51 \)[/tex] can be written as [tex]\( 50 + 1 \)[/tex] and [tex]\( 49 \)[/tex] can be written as [tex]\( 50 - 1 \)[/tex]. We can use the difference of squares formula here as well:
[tex]\[ (50 + 1)(50 - 1) = 50^2 - 1^2 \][/tex]
Let's calculate each part:
[tex]\[ 50^2 = 2500 \][/tex]
[tex]\[ 1^2 = 1 \][/tex]
Now, substituting back into the formula:
[tex]\[ 2500 - 1 = 2499 \][/tex]
So, the simplified result of [tex]\( 51 \times 49 \)[/tex] is:
[tex]\[ 2499 \][/tex]
### Summary of Results
- For part a, [tex]\( 25^2 - 15^2 = 400 \)[/tex]
- For part e, [tex]\( 51 \times 49 = 2499 \)[/tex]
