Simplify (5b^2+7a^2)(-5b^2+7a^2)

$(5 b^{2} + 7 a^{2}) (- 5 b^{2} + 7 a^{2})$

Step 1

Expand $(5 b^{2} + 7 a^{2}) (- 5 b^{2} + 7 a^{2})$ using the FOIL Method.

Apply the distributive property.

$5 b^{2} (- 5 b^{2} + 7 a^{2}) + 7 a^{2} (- 5 b^{2} + 7 a^{2})$

Apply the distributive property.

$5 b^{2} (- 5 b^{2}) + 5 b^{2} (7 a^{2}) + 7 a^{2} (- 5 b^{2} + 7 a^{2})$

Apply the distributive property.

$5 b^{2} (- 5 b^{2}) + 5 b^{2} (7 a^{2}) + 7 a^{2} (- 5 b^{2}) + 7 a^{2} (7 a^{2})$

Step 2

Simplify terms.

Combine the opposite terms in $5 b^{2} (- 5 b^{2}) + 5 b^{2} (7 a^{2}) + 7 a^{2} (- 5 b^{2}) + 7 a^{2} (7 a^{2})$ .

Reorder the factors in the terms $5 b^{2} (7 a^{2})$ and $7 a^{2} (- 5 b^{2})$ .

$5 b^{2} (- 5 b^{2}) + 5 \cdot 7 a^{2} b^{2} - 5 \cdot 7 a^{2} b^{2} + 7 a^{2} (7 a^{2})$

Subtract $5 \cdot 7 a^{2} b^{2}$ from $5 \cdot 7 a^{2} b^{2}$ .

$5 b^{2} (- 5 b^{2}) + 0 + 7 a^{2} (7 a^{2})$

Add $5 b^{2} (- 5 b^{2})$ and $0$ .

$5 b^{2} (- 5 b^{2}) + 7 a^{2} (7 a^{2})$

Simplify each term.

Rewrite using the commutative property of multiplication.

$5 \cdot - 5 b^{2} b^{2} + 7 a^{2} (7 a^{2})$

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

$5 \cdot - 5 (b^{2} b^{2}) + 7 a^{2} (7 a^{2})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 \cdot - 5 b^{2 + 2} + 7 a^{2} (7 a^{2})$

Add $2$ and $2$ .

$5 \cdot - 5 b^{4} + 7 a^{2} (7 a^{2})$

Multiply $5$ by $- 5$ .

$- 25 b^{4} + 7 a^{2} (7 a^{2})$

Rewrite using the commutative property of multiplication.

$- 25 b^{4} + 7 \cdot 7 a^{2} a^{2}$

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

$- 25 b^{4} + 7 \cdot 7 (a^{2} a^{2})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 25 b^{4} + 7 \cdot 7 a^{2 + 2}$

Add $2$ and $2$ .

$- 25 b^{4} + 7 \cdot 7 a^{4}$

Multiply $7$ by $7$ .

$- 25 b^{4} + 49 a^{4}$