Question

select the correct answer.
which expression simplifies to $4\sqrt{13}$?
\bigcirc $\sqrt{17}$
\bigcirc $\sqrt{52}$
\bigcirc $\sqrt{208}$
\bigcirc $\sqrt{29}$

Explanation:

Step1: Recall the property of square roots

We know that \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) for \( a\geq0,b\geq0 \). Also, if we have \( c\sqrt{d} \), then \( (c\sqrt{d})^2 = c^{2}\times d \). So to find which square root simplifies to \( 4\sqrt{13} \), we can square \( 4\sqrt{13} \) and see which option's radicand matches the result.

Step2: Square the expression \( 4\sqrt{13} \)

Using the formula \( (ab)^2=a^{2}b^{2} \), where \( a = 4 \) and \( b=\sqrt{13} \), we get:
\( (4\sqrt{13})^{2}=4^{2}\times(\sqrt{13})^{2} \)
\( = 16\times13 \)
\( = 208 \)

So the expression \( \sqrt{208} \) is equivalent to \( 4\sqrt{13} \) because \( \sqrt{208}=\sqrt{16\times13}=\sqrt{16}\times\sqrt{13} = 4\sqrt{13} \). We can also check the other options:

• For \( \sqrt{17} \), 17 is a prime number, so it cannot be simplified to \( 4\sqrt{13} \).
• For \( \sqrt{52} \), \( \sqrt{52}=\sqrt{4\times13}=2\sqrt{13}

eq4\sqrt{13} \).

• For \( \sqrt{29} \), 29 is a prime number, so it cannot be simplified to \( 4\sqrt{13} \).

Answer:

\( \boldsymbol{\sqrt{208}} \) (the option with \( \sqrt{208} \))