Question

1. √(2⁵ · 13³)

Explanation:

Step1: Simplify exponents inside sqrt

Rewrite \(2^5\) as \(2^4 \cdot 2\) and \(13^3\) as \(13^2 \cdot 13\). So, \(\sqrt{2^5 \cdot 13^3} = \sqrt{2^4 \cdot 2 \cdot 13^2 \cdot 13}\).

Step2: Apply square - root property

Use \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a,b\geq0\)) and \(\sqrt{x^2}=x\) (\(x\geq0\)). Then \(\sqrt{2^4 \cdot 2 \cdot 13^2 \cdot 13}=\sqrt{2^4}\cdot\sqrt{13^2}\cdot\sqrt{2\cdot13}\).
Since \(\sqrt{2^4} = 2^2=4\) and \(\sqrt{13^2}=13\), we have \(4\times13\times\sqrt{26}\).

Step3: Calculate the product of constants

Calculate \(4\times13 = 52\). So the expression simplifies to \(52\sqrt{26}\). If we want a decimal approximation, \(\sqrt{26}\approx5.099\), then \(52\times5.099\approx265.15\). But in exact form, it is \(52\sqrt{26}\).

Answer:

The simplified form of \(\sqrt{2^5\cdot13^3}\) is \(52\sqrt{26}\) (or approximately \(265.15\) if a decimal approximation is needed).