if a cubed minus B cubed equals 120 and a minus B equals three, then what is a times B

We are given the equations:

1. \( a^3 - b^3 = 120 \)
2. \( a - b = 3 \)

We will use the factorization of the difference of cubes formula:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Substituting \( a - b = 3 \) into the equation:

\[
120 = 3(a^2 + ab + b^2)
\]

Dividing both sides by 3:

\[
a^2 + ab + b^2 = 40
\]

Next, let's square the equation \( a - b = 3 \):

\[
(a - b)^2 = a^2 - 2ab + b^2 = 9
\]

We now have two equations:

1. \( a^2 + ab + b^2 = 40 \)
2. \( a^2 - 2ab + b^2 = 9 \)

Subtract the second equation from the first:

\[
(a^2 + ab + b^2) - (a^2 - 2ab + b^2) = 40 - 9
\]

Simplifying:

\[
3ab = 31
\]

So,

\[
ab = \frac{31}{3}
\]

Thus, \( a \times b = \frac{31}{3} \).

Would you like more details or have any questions?

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Here are 5 related questions for practice:

1. How do you factor \( a^3 - b^3 \)?
2. What is the value of \( a + b \) if \( a - b = 3 \)?
3. How can you find the individual values of \( a \) and \( b \) given \( ab \) and \( a - b \)?
4. How would you solve \( a^2 + b^2 \) if \( ab \) is known?
5. What happens if \( a^3 + b^3 = 120 \) instead?

**Tip:** Always consider factorization techniques for powers and roots, as they simplify many algebraic equations.

If a cubed minus B cubed equals 120 and a minus B equals three, then what is a times B?

Explore how to solve the algebraic problem where a^3 - b^3 equals 120 and a - b equals 3 to find a × b. This solution involves the difference of cubes formula and algebraic manipulation. Suitable for students in Grades 8-10.

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