Question

How do you simplify \[\sqrt {512} \]?

Answer 1

Hint: Here, we will first rewrite the number 512 as the product of its prime factors. Then, we will simplify it using the product rule of square roots. Finally, we will simplify the radicals to find the required value.

Formula Used:
We will use the following formulas:
1. \[{a^b} \times {a^c} = {a^{b + c}}\]
2. The product rule of square roots states that \[\sqrt {A \times B} = \sqrt A \sqrt B \], where A and B are real numbers.

Complete step-by-step solution:
First, we will rewrite 512 as the product of its prime factors.
We know that the prime factors of 512 are 2, 2, 2, 2, 2, 2, 2, 2, and 2.
Therefore, we can rewrite the number 512 as
\[512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
Rule of exponents: If two or more numbers with the same base and different exponents are multiplied, the product can be written as \[{a^b} \times {a^c} = {a^{b + c}}\].
Therefore, using the rule of exponents \[{a^b} \times {a^c} = {a^{b + c}}\], we can rewrite 512 as
\[\begin{array}{l} \Rightarrow 512 = {2^1} \times {2^1} \times {2^1} \times {2^1} \times {2^1} \times {2^1} \times {2^1} \times {2^1} \times {2^1}\\ \Rightarrow 512 = {2^8} \times 2\end{array}\]
Substituting \[512 = {2^8} \times 2\] in the expression \[\sqrt {512} \], we get
\[\sqrt {512} = \sqrt {{2^8} \times 2} \]
The product rule of square roots states that \[\sqrt {A \times B} = \sqrt A \sqrt B \].
Applying the product rule of square roots in the expression, we get
\[ \Rightarrow \sqrt {512} = \sqrt {{2^8}} \sqrt 2 \]
The square root of a number \[{a^n}\] can be written as \[{a^{\dfrac{n}{2}}}\].
Simplifying the radical in the expression, we get
\[ \Rightarrow \sqrt {512} = {2^{\dfrac{8}{2}}}\sqrt 2 \]
Simplifying the exponent, we get
\[ \Rightarrow \sqrt {512} = {2^4}\sqrt 2 \]
The number 2 raised to the power 4 is equal to 16.
Therefore, we get
\[ \Rightarrow \sqrt {512} = 16\sqrt 2 \]

Therefore, we have simplified the expression \[\sqrt {512} \] as \[16\sqrt 2 \].

Note:
We have used the term ‘prime factor’ in the solution. A prime factor is a factor that is a prime number. Prime numbers are the numbers that have only two factors, 1 and the number itself. For example, 2 is the product of 2 and 1. Therefore, the factors of 2 are 2 and 1. Thus, 2 is a prime number. We need to keep in mind that the square root of a number is a factor which when multiplied by itself gives the original number.