(a^(1/2)+b^(1/2))^3 * (a^(1/2) - b^(1/2))^3

Expand First Term: First, let's expand (a^(1/2)+b^(1/2))^3 using the binomial theorem. (a^(1/2)+b^(1/2))^3 = a^(3/2) + 3a * b^(1/2) + 3a^(1/2) * b + b^(3/2)Expand Second Term: Now, let's expand (a^(1/2) - b^(1/2))^3 using the binomial theorem. (a^(1/2) - b^(1/2))^3 = a^(3/2) - 3a * b^(1/2) + 3a^(1/2) * b - b^(3/2)Multiply Expanded Forms: Next, we multiply the two expanded forms together. (a^(3/2) + 3a * b^(1/2) + 3a^(1/2) * b + b^(3/2)) * (a^(3/2) - 3a * b^(1/2) + 3a^(1/2) * b - b^(3/2))Apply Difference of Squares: We notice that this is a product of a sum and a difference, which resembles the pattern (x + y)(x - y) = x^2 - y^2. So, we can simplify the expression to (a^(3/2))^2 - (b^(3/2))^2.Calculate Final Result: Now, let's calculate (a^(3/2))^2 and (b^(3/2))^2. (a^(3/2))^2 = a^(3) and (b^(3/2))^2 = b^(3)Subtract the two values to get the final result. a^(3) - b^(3)

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$(a^{\frac{1}{2}}+b^{\frac{1}{2}})^3 \cdot (a^{\frac{1}{2}} - b^{\frac{1}{2}})^3$

View step-by-step help

Home

Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

(a^{\frac{1}{2}}+b^{\frac{1}{2}})^3 \cdot (a^{\frac{1}{2}} - b^{\frac{1}{2}})^3

Expand First Term: First, let's expand $(a^{\frac{1}{2}}+b^{\frac{1}{2}})^3$ using the binomial theorem. $\newline$ $(a^{\frac{1}{2}}+b^{\frac{1}{2}})^3 = a^{\frac{3}{2}} + 3a \cdot b^{\frac{1}{2}} + 3a^{\frac{1}{2}} \cdot b + b^{\frac{3}{2}}$
Expand Second Term: Now, let's expand $(a^{\frac{1}{2}} - b^{\frac{1}{2}})^3$ using the binomial theorem. $\newline$ $(a^{\frac{1}{2}} - b^{\frac{1}{2}})^3 = a^{\frac{3}{2}} - 3a \cdot b^{\frac{1}{2}} + 3a^{\frac{1}{2}} \cdot b - b^{\frac{3}{2}}$
Multiply Expanded Forms: Next, we multiply the two expanded forms together. $\newline$ $(a^{\frac{3}{2}} + 3a \cdot b^{\frac{1}{2}} + 3a^{\frac{1}{2}} \cdot b + b^{\frac{3}{2}}) \cdot (a^{\frac{3}{2}} - 3a \cdot b^{\frac{1}{2}} + 3a^{\frac{1}{2}} \cdot b - b^{\frac{3}{2}})$
Apply Difference of Squares: We notice that this is a product of a sum and a difference, which resembles the pattern $(x + y)(x - y) = x^2 - y^2$ . So, we can simplify the expression to $(a^{(3/2)})^2 - (b^{(3/2)})^2$ .
Calculate Final Result: Now, let's calculate $(a^{\frac{3}{2}})^2$ and $(b^{\frac{3}{2}})^2$ . $(a^{\frac{3}{2}})^2 = a^{3}$ and $(b^{\frac{3}{2}})^2 = b^{3}$
Calculate Final Result: Now, let's calculate $(a^{\frac{3}{2}})^2$ and $(b^{\frac{3}{2}})^2$ . $(a^{\frac{3}{2}})^2 = a^{3}$ and $(b^{\frac{3}{2}})^2 = b^{3}$ Subtract the two values to get the final result. $a^{3} - b^{3}$