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Simplify the following expression:

[tex]\[ a^4 - \left(a^2 - 3b^2\right)^2 \][/tex]

Sagot :

GinnyAnsweridn
Let's simplify the expression [tex]\( a^4 - (a^2 - 3b^2)^2 \)[/tex] step-by-step.

1. Expand the squared term:

[tex]\[ (a^2 - 3b^2)^2 \][/tex]

To expand this, we use the formula for the square of a binomial, [tex]\((x - y)^2 = x^2 - 2xy + y^2\)[/tex]:

[tex]\[ (a^2 - 3b^2)^2 = (a^2)^2 - 2(a^2)(3b^2) + (3b^2)^2 \][/tex]

Calculating each term:

[tex]\[ (a^2)^2 = a^4 \][/tex]
[tex]\[ 2(a^2)(3b^2) = 6a^2b^2 \][/tex]
[tex]\[ (3b^2)^2 = 9b^2 \][/tex]

So,

[tex]\[ (a^2 - 3b^2)^2 = a^4 - 6a^2b^2 + 9b^4 \][/tex]

2. Substitute the expanded term back into the original expression:

[tex]\[ a^4 - (a^2 - 3b^2)^2 = a^4 - \left(a^4 - 6a^2b^2 + 9b^4\right) \][/tex]

3. Distribute the negative sign:

[tex]\[ a^4 - (a^4 - 6a^2b^2 + 9b^4) = a^4 - a^4 + 6a^2b^2 - 9b^4 \][/tex]

4. Combine like terms:

[tex]\[ a^4 - a^4 + 6a^2b^2 - 9b^4 = 6a^2b^2 - 9b^4 \][/tex]

Therefore, the simplified form of the expression [tex]\( a^4 - (a^2 - 3b^2)^2 \)[/tex] is:

[tex]\[ 6a^2b^2 - 9b^4 \][/tex]
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