What is the square root of 1458?

Steps to Solve

1. Factor the number Start by finding the prime factors of 1458. We can divide by 2, the first prime number: $$ 1458 \div 2 = 729 $$ Next, we can factor 729: $$ 729 = 3 \times 243 $$ Continuing with 243: $$ 243 = 3 \times 81 $$ Next, factor 81: $$ 81 = 3 \times 27 $$ And then factor 27: $$ 27 = 3 \times 9 $$ Finally, factor 9: $$ 9 = 3 \times 3 $$ So, the complete factorization of 1458 is: $$ 1458 = 2 \times 3^6 $$

2. Use the property of square roots The property of square roots states that: $$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $$ We will apply this to our factorization: $$ \sqrt{1458} = \sqrt{2 \times 3^6} $$

3. Simplify using square root properties Now simplify: $$ \sqrt{1458} = \sqrt{2} \times \sqrt{3^6} $$ Since $3^6$ is a perfect square, we can simplify this further: $$ \sqrt{3^6} = 3^{6/2} = 3^3 = 27 $$ Thus: $$ \sqrt{1458} = 27 \times \sqrt{2} $$

4. Final result The square root of 1458 is: $$ \sqrt{1458} = 27\sqrt{2} $$