Factorise fully. 1.2.1. $$ \quad\left(4 x^{6}-y^{4}\right) $$

Question

UpStudy AI · Accepted Answer

We start with the expression

$$
4x^6 - y^4.
$$

**Step 1. Recognise the difference of squares.**

Notice that

$$
4x^6 = (2x^3)^2 \quad \text{and} \quad y^4 = (y^2)^2.
$$

Thus, we can write the expression as

$$
(2x^3)^2 - (y^2)^2.
$$

**Step 2. Apply the difference of squares factorisation formula.**

Recall the formula

$$
a^2 - b^2 = (a - b)(a + b).
$$

Let $$ a = 2x^3 $$ and $$ b = y^2 $$. Applying the formula gives

$$
(2x^3)^2 - (y^2)^2 = (2x^3 - y^2)(2x^3 + y^2).
$$

**Step 3. Conclude the factorisation.**

The fully factorised form of the expression is

$$
(2x^3 - y^2)(2x^3 + y^2).
$$
The expression $$ 4x^6 - y^4 $$ can be fully factored as $$ (2x^3 - y^2)(2x^3 + y^2) $$.

Factorise fully. 1.2.1. \( \quad\left(4 x^{6}-y^{4}\right) \)