Final answer:
The product of the first terms using the FOIL method for the multiplication problem (2√7 + 9√5) (4√7 - 8√5) is 56, achieved by multiplying 2√7 by 4√7.
Step-by-step explanation:
To find the product of the first terms using the FOIL method for the multiplication problem (2√7 + 9√5) (4√7 - 8√5), we multiply the 'first' terms from each binomial.
The first term in the first binomial is 2√7, and the first term in the second binomial is 4√7. When we multiply these together, we use the property that √x^{2} = √x. So, if we consider √7 as 7^{1/2}, when we multiply 2√7 by 4√7, we multiply the coefficients (2 and 4) and the square root of 7 by itself, which is just 7.
The multiplication of the coefficients 2 and 4 gives us 8. The square root of 7 multiplied by itself gives us 7. Multiplying these results together (8 and 7) following the rule that when two positive numbers multiply, the answer has a positive sign, we get the product 56.
Therefore, the product of the first terms in the given multiplication problem is 56.