Question

complete the procedure to find the equivalent expression of 252^{\frac{1}{2}}. simplify. sqrt{4cdot7cdot9}, sqrt{4}cdotsqrt{7}cdotsqrt{9}, sqrt{252}, 252^{\frac{1}{2}}

Explanation:

Step1: Recall square - root properties

We know that $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$ and $\sqrt{a^2}=a$ ($a\geq0$). First, simplify $\sqrt{4}\cdot\sqrt{7}\cdot\sqrt{9}$. Since $\sqrt{4} = 2$, $\sqrt{9}=3$, then $\sqrt{4}\cdot\sqrt{7}\cdot\sqrt{9}=2\cdot\sqrt{7}\cdot3 = 6\sqrt{7}$.

Step2: Simplify $\sqrt{4\cdot7\cdot9}$

Calculate $4\times7\times9=252$, so $\sqrt{4\cdot7\cdot9}=\sqrt{252}$.

Step3: Simplify $\sqrt{252}$

Factorize 252: $252=2^2\times3^2\times7$. Then $\sqrt{252}=\sqrt{2^2\times3^2\times7}=2\times3\sqrt{7}=6\sqrt{7}$.

Step4: Simplify $252^{\frac{1}{2}}$

By the definition of fractional exponents, $a^{\frac{1}{n}}=\sqrt[n]{a}$. When $n = 2$, $252^{\frac{1}{2}}=\sqrt{252}=6\sqrt{7}$.

Answer:

$6\sqrt{7}$