1. **Statement of the problem:** Calculate the value of $$\sqrt{7} \times (-\sqrt{7})$$ and compare it to $$ (\sqrt{7})^{100} \times (-\sqrt{7})^{100}$$. 2. **Recall the properties and formulas:** - The square root of a number squared is the number itself: $$ (\sqrt{a})^2 = a $$. - Multiplying square roots: $$ \sqrt{a} \times \sqrt{a} = a $$. - Powers of products: $$ (xy)^n = x^n y^n $$. - Negative powers and signs: $$ (-x)^n = (-1)^n x^n $$. 3. **Calculate the left side:** $$ \sqrt{7} \times (-\sqrt{7}) = - (\sqrt{7} \times \sqrt{7}) = -7 $$ 4. **Calculate the right side:** $$ (\sqrt{7})^{100} \times (-\sqrt{7})^{100} = (\sqrt{7})^{100} \times ((-1)^{100} (\sqrt{7})^{100}) = (\sqrt{7})^{100} \times (1) \times (\sqrt{7})^{100} = (\sqrt{7})^{200} $$ Since $$ (\sqrt{7})^{200} = (7^{1/2})^{200} = 7^{100} $$ 5. **Compare both sides:** Left side = $$ -7 $$ Right side = $$ 7^{100} $$ These are not equal unless $$ 7^{100} = -7 $$ which is false. **Final answer:** $$ \sqrt{7} \times (-\sqrt{7}) = -7 $$ **Slug:** sqrt7 product **Subject:** algebra "desmos": {"latex": "y = -7", "features": {"intercepts": true, "extrema": true}}, "q_count": 5