Embibe Experts Solutions for Chapter: Real Numbers, Exercise 1: Exercise

The number ${\left(\sqrt{3}+\sqrt{5}\right)}^2$ is

The product of a non-zero rational and irrational number is

Parul and Nupur both study in class $\mathrm{X}$. Today, they were taught about the Fundamental Theorem of Arithmetic which states that, "Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur".

To explain the Fundamental Theorem of Arithmetic, their teacher recalled the prime factorisation of a number by the 'factor tree method'. 

Parul became interested in the factor tree method and she asked Nupur to help to complete the following factor tree.

Find the missing numbers.

Explain why $11\times 13\times 15\times 17+17$ is a composite number?

Explain why $17\times 5\times 11\times 3\times 2+2\times 11$ is a composite number?

Multiply $3$$\sqrt{12}$ and$3\sqrt{18}$

Prove that $7-\frac{2}{3}\sqrt{5}$ is irrational.

Solve: $6\sqrt{15}\times 4\sqrt{3}$