Question

1. \\(\sqrt3{81} \cdot \sqrt3{45}\\) 38. \\(\sqrt{\frac{845}{5}}\\)

Explanation:

Problem 37:

Step1: Recall the property of cube roots

For real numbers \(a\) and \(b\), \(\sqrt[3]{a}\cdot\sqrt[3]{b}=\sqrt[3]{a\cdot b}\). So we can multiply the radicands first.
\(\sqrt[3]{81}\cdot\sqrt[3]{45}=\sqrt[3]{81\times45}\)

Step2: Factor the product inside the cube root

Factor \(81 = 3^4\) and \(45=3^2\times5\). Then \(81\times45 = 3^4\times3^2\times5=3^{4 + 2}\times5=3^6\times5\)
So \(\sqrt[3]{3^6\times5}\)

Step3: Simplify the cube root

We know that \(\sqrt[3]{x^n}=x^{\frac{n}{3}}\) for \(x\geq0\). For \(x = 3^6\), \(n = 6\), so \(\sqrt[3]{3^6}=3^{\frac{6}{3}}=3^2 = 9\)
Thus, \(\sqrt[3]{3^6\times5}=9\sqrt[3]{5}\)

Problem 38:

Step1: Simplify the fraction inside the square root

First, simplify \(\frac{845}{5}\). Divide 845 by 5: \(845\div5 = 169\)
So \(\sqrt{\frac{845}{5}}=\sqrt{169}\)

Step2: Evaluate the square root

We know that \(13\times13 = 169\), so \(\sqrt{169}=13\)

Problem 37 Answer: \(9\sqrt[3]{5}\)

Problem 38 Answer: \(13\)

Answer:

Step1: Simplify the fraction inside the square root

First, simplify \(\frac{845}{5}\). Divide 845 by 5: \(845\div5 = 169\)
So \(\sqrt{\frac{845}{5}}=\sqrt{169}\)

Step2: Evaluate the square root

We know that \(13\times13 = 169\), so \(\sqrt{169}=13\)

Problem 37 Answer: \(9\sqrt[3]{5}\)

Problem 38 Answer: \(13\)