answer:The null space of the given matrix is spanned by the following basis vectors: begin{align*} v_1 &= (1, 0, -9, 0) v_2 &= (0, 1, 0, -9) v_3 &= (0, 0, 0, 1) end{align*} These vectors are already rounded to three decimal places, and they form a basis since they are linearly independent and span the null space.

answer:Hormones and neurotransmitters both facilitate communication within the body, but their mechanisms of action differ significantly. Hormones are released by endocrine glands into the bloodstream and travel throughout the body until they reach target cells with specific receptors. They induce changes in cellular activity, often over longer periods. Neurotransmitters, on the other hand, are released by neurons at synapses and act locally on neighboring cells. They bind to receptors on the postsynaptic membrane, causing immediate changes in electrical potential or chemical signaling within the target cell.

answer:Step 1: Isolate the radical terms on one side of the equation. sqrt{frac{9x}{2}-frac{21}{2}}+sqrt{frac{25x}{2}+frac{17}{2}}=9 (sqrt{frac{9x}{2}-frac{21}{2}}+sqrt{frac{25x}{2}+frac{17}{2}})^2=9^2 frac{9x}{2}-frac{21}{2}+frac{25x}{2}+frac{17}{2}+2sqrt{(frac{9x}{2}-frac{21}{2})(frac{25x}{2}+frac{17}{2})}=81 17x+2sqrt{frac{225x^2}{4}-frac{81x}{4}-frac{351}{4}}=81 17x+2sqrt{frac{225x^2-81x-351}{4}}=81 17x+2sqrt{frac{(15x-49)(15x+7)}{4}}=81 17x+sqrt{(15x-49)(15x+7)}=40.5 Step 2: Square both sides of the equation. (17x+sqrt{(15x-49)(15x+7)})^2=40.5^2 289x^2+140xsqrt{(15x-49)(15x+7)}+15x-49=1640.25 289x^2+140xsqrt{(15x-49)(15x+7)}=1640.25-15x+49 289x^2+140xsqrt{(15x-49)(15x+7)}=1674.25-15x Step 3: Solve for x. 140xsqrt{(15x-49)(15x+7)}=1674.25-15x-289x^2 (140x)^2[(15x-49)(15x+7)]=(1674.25-15x-289x^2)^2 19600x^2(225x^2-351x-343)=2801.0625-67.5x+225x^2-255x^3+841x^4 841x^4-255x^3+225x^2-67.5x+2801.0625=19600x^2(225x^2-351x-343) 841x^4-255x^3+225x^2-67.5x+2801.0625=44100x^4-693600x^3-670200x^2 -43659x^4+693855x^3+670425x^2-67.5x+2801.0625=0 -43659x^4+693855x^3+670425x^2-67.5x+2801.0625=0 xapprox 0.0156 Therefore, the only real solution to the equation is approximately xapprox 0.0156. The answer is left{frac{1}{64} left(1225-9 sqrt{12801}right)right}

answer:The equation shows that the total mass of the reactants (4 grams of hydrogen and 32 grams of oxygen) is equal to the total mass of the product (36 grams of water). This demonstrates that mass is neither created nor destroyed in the reaction, satisfying the principle of mass balance.