Question

mathematical concepts evolve, your mathematical knowledge evolves as well. previous knowledge provides the necessary skills to acquire new learning. if necessary, review the simplifying and combining radicals skill builder, to strengthen your understanding of this topic. homework help simplify each of the following radical expressions.
a. $sqrt{12}+sqrt{\frac{3}{4}}-sqrt{27}$
b. $(2sqrt{b}-sqrt{a})(sqrt{b}+3sqrt{a})$
c. $(sqrt{2}-sqrt{3})^2$
d. $sqrt4{x^{4}y^{9}}$

Explanation:

Step1: Simplify radicals in a

Simplify each radical term.
$\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}$, $\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$, $\sqrt{27}=\sqrt{9\times3}=3\sqrt{3}$.
So $\sqrt{12}+\sqrt{\frac{3}{4}}-\sqrt{27}=2\sqrt{3}+\frac{\sqrt{3}}{2}-3\sqrt{3}$.

Step2: Combine like - terms in a

$2\sqrt{3}+\frac{\sqrt{3}}{2}-3\sqrt{3}=(2 + \frac{1}{2}-3)\sqrt{3}=(\frac{4 + 1-6}{2})\sqrt{3}=-\frac{1}{2}\sqrt{3}$.

Step3: Expand the product in b

Use FOIL method: $(2\sqrt{b}-\sqrt{a})(\sqrt{b}+3\sqrt{a})=2\sqrt{b}\times\sqrt{b}+2\sqrt{b}\times3\sqrt{a}-\sqrt{a}\times\sqrt{b}-\sqrt{a}\times3\sqrt{a}$.
$=2b + 6\sqrt{ab}-\sqrt{ab}-3a=2b-3a + 5\sqrt{ab}$.

Step4: Expand the square in c

Use the formula $(m - n)^2=m^2-2mn + n^2$. Here $m=\sqrt{2}$ and $n=\sqrt{3}$.
$(\sqrt{2}-\sqrt{3})^2=(\sqrt{2})^2-2\sqrt{2}\times\sqrt{3}+(\sqrt{3})^2=2-2\sqrt{6}+3=5 - 2\sqrt{6}$.

Step5: Simplify the fourth - root in d

$\sqrt[4]{x^{4}y^{9}}=\sqrt[4]{x^{4}y^{8}\times y}=|x|y^{2}\sqrt[4]{y}$.

Answer:

a. $-\frac{1}{2}\sqrt{3}$
b. $2b-3a + 5\sqrt{ab}$
c. $5 - 2\sqrt{6}$
d. $|x|y^{2}\sqrt[4]{y}$