Factor 27t^(3)-125v^(3) completely. Answer:

Identify Form: Identify the form of the expression. The expression 27t^(3)-125v^(3) is a difference of cubes since 27 and 125 are both perfect cubes (3^3 and 5^3, respectively), and t^3 and v^3 are cubes of variables.Write Formula: Write down the formula for factoring a difference of cubes. The formula is a^3 - b^3 = (a - b)(a^2 + ab + b^2).Identify a, b: Identify 'a' and 'b' in the expression 27t^(3)-125v^(3). Here, a = 3t and b = 5v because (3t)^3 = 27t^3 and (5v)^3 = 125v^3.Apply Formula: Apply the difference of cubes formula to the expression. Using a = 3t and b = 5v, we get (3t - 5v)((3t)^2 + (3t)(5v) + (5v)^2).Simplify Inside: Simplify the expression inside the parentheses. Calculate (3t)^2, (3t)(5v), and (5v)^2 to get 9t^2, 15tv, and 25v^2, respectively.Write Fully Factored: Write the fully factored form of the expression. The factored form is (3t - 5v)(9t^2 + 15tv + 25v^2).

Home

Math Problems

Precalculus

Operations with rational exponents

Full solution

Q. Factor

27 t^{3}-125 v^{3}

completely.

\newline

Answer:

Identify Form: Identify the form of the expression. The expression $27t^{3}-125v^{3}$ is a difference of cubes since $27$ and $125$ are both perfect cubes ( $3^{3}$ and $5^{3}$ , respectively), and $t^{3}$ and $v^{3}$ are cubes of variables.
Write Formula: Write down the formula for factoring a difference of cubes. The formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .
Identify $a$ , $b$ : Identify ' $a$ ' and ' $b$ ' in the expression $27t^{3}-125v^{3}$ . Here, $a = 3t$ and $b = 5v$ because $(3t)^3 = 27t^3$ and $(5v)^3 = 125v^3$ .
Apply Formula: Apply the difference of cubes formula to the expression. Using $a = 3t$ and $b = 5v$ , we get $(3t - 5v)((3t)^2 + (3t)(5v) + (5v)^2)$ .
Simplify Inside: Simplify the expression inside the parentheses. Calculate $(3t)^2$ , $(3t)(5v)$ , and $(5v)^2$ to get $9t^2$ , $15tv$ , and $25v^2$ , respectively.
Write Fully Factored: Write the fully factored form of the expression. The factored form is $(3t - 5v)(9t^2 + 15tv + 25v^2)$ .