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Factorise $48a^4-243b^4$

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Simplify rational expressions

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Q. Factorise

48a^4-243b^4

Recognize as difference of squares: Recognize the expression as a difference of squares. The expression $48a^4 - 243b^4$ can be seen as a difference of two squares since both $48a^4$ and $243b^4$ are perfect squares. The difference of squares formula is $A^2 - B^2 = (A + B)(A - B)$ .
Express as squares: Express each term as a square. $48a^4$ is $(4a^2)^2$ and $243b^4$ is $(3b^2)^2$ . So, we can rewrite the expression as $(4a^2)^2 - (3b^2)^2$ .
Apply formula: Apply the difference of squares formula. $\newline$ Using the formula from Step $1$ , we can factor the expression as follows: $\newline$ $(4a^2)^2 - (3b^2)^2 = (4a^2 + 3b^2)(4a^2 - 3b^2)$ .
Check for further factorization: Check for further factorization. $\newline$ The terms $(4a^2 + 3b^2)$ and $(4a^2 - 3b^2)$ cannot be factored further using real numbers since they are not differences or sums of squares or any other factorable form. Therefore, the factorization is complete.

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