Question Simplify the expression $(v-d)(v+d)(v^2-d^2)$ .

Studdy Solution

STEP 1

Assumptions
1. We need to simplify the expression $(v-d)(v+d)(v^2-d^2)$ .
2. We will use the distributive property and the difference of squares formula.

STEP 2

Recognize that $(v-d)(v+d)$ and $(v^2-d^2)$ are both instances of the difference of squares formula, which states that $a^2 - b^2 = (a-b)(a+b)$ .

STEP 3

Apply the difference of squares formula to $(v-d)(v+d)$ , which simplifies to $v^2 - d^2$ .

STEP 4

Now we have $(v^2 - d^2)(v^2 - d^2)$ , which is the product of two identical expressions.

STEP 5

Use the power rule of exponents to simplify the expression further. The power rule states that $(a^b)^c = a^{b \cdot c}$ when multiplying powers with the same base.

STEP 6

Apply the power rule to our expression, recognizing that $(v^2 - d^2)$ is being squared.

STEP 7

Simplify the expression using the power rule.
$(v^2 - d^2)^2 = (v^2)^2 - 2(v^2)(d^2) + (d^2)^2$

STEP 8

Calculate each term separately.
$(v^2)^2 = v^4$ $(d^2)^2 = d^4$ $2(v^2)(d^2) = 2v^2d^2$

STEP 9

Combine the terms to get the final simplified expression.
$(v^2 - d^2)^2 = v^4 - 2v^2d^2 + d^4$
The simplified expression is $v^4 - 2v^2d^2 + d^4$ .

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