Question

use the quotient rule to simplify. assume all variables represent non - negative values.
\sqrt{\frac{7m^{5}}{28m}}
\sqrt{\frac{7m^{5}}{28m}} = \square
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)
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question 8 of 20 back

Explanation:

Step1: Simplify the fraction inside the square root

First, simplify the fraction \(\frac{7m^5}{28m}\). We can divide the coefficients and use the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n} \) for the variable \( m \).

For the coefficients: \(\frac{7}{28}=\frac{1}{4}\)

For the variable \( m \): \(\frac{m^5}{m}=m^{5 - 1}=m^4\)

So, \(\frac{7m^5}{28m}=\frac{1}{4}m^4\)

Step2: Apply the square root to the simplified fraction

Now we have \(\sqrt{\frac{1}{4}m^4}\). We can use the property of square roots \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (where \( a\geq0 \) and \( b\geq0 \)) and also \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) (where \( a\geq0 \) and \( b > 0 \)).

First, \(\sqrt{\frac{1}{4}m^4}=\sqrt{\frac{1}{4}}\cdot\sqrt{m^4}\)

We know that \(\sqrt{\frac{1}{4}}=\frac{1}{2}\) and \(\sqrt{m^4}=m^2\) (since \( m \) is non - negative, \(\sqrt{m^{2k}}=m^k\) for \( k\in\mathbb{N}\))

So, \(\sqrt{\frac{1}{4}}\cdot\sqrt{m^4}=\frac{1}{2}\cdot m^2=\frac{m^2}{2}\)

Answer:

\(\frac{m^2}{2}\)