1. **State the problem:** We need to find the exponents in the expression $(x^{a} + 2)(x^{b} + 8) = x^{8} + 10x^{4} + 16$ such that the equation is true. 2. **Recall the formula:** When multiplying binomials, use the distributive property: $$ (x^{a} + 2)(x^{b} + 8) = x^{a} \cdot x^{b} + 8x^{a} + 2x^{b} + 16 $$ 3. **Simplify the terms:** $$ x^{a+b} + 8x^{a} + 2x^{b} + 16 $$ 4. **Match terms with the right side:** The right side is: $$ x^{8} + 10x^{4} + 16 $$ So, we must have: - The highest power term: $x^{a+b} = x^{8}$ so $a + b = 8$ - The middle terms combined: $8x^{a} + 2x^{b} = 10x^{4}$ - The constant term: 16 matches 16 5. **Analyze the middle terms:** Since the middle term is $10x^{4}$, the powers of $x$ in both terms must be 4 to combine like terms: $a = 4$ and $b = 4$ 6. **Check the sum of exponents:** $a + b = 4 + 4 = 8$, which matches the highest power term. 7. **Verify the entire expression:** $$ (x^{4} + 2)(x^{4} + 8) = x^{8} + 8x^{4} + 2x^{4} + 16 = x^{8} + 10x^{4} + 16 $$ This confirms the exponents are both 4. **Final answer:** $$ (x^{4} + 2)(x^{4} + 8) = x^{8} + 10x^{4} + 16 $$