To simplify the expression [tex]\( 8 \sqrt{112 b} \)[/tex], let's follow the step-by-step process below:
1. Factorize 112: The number 112 can be broken down into its prime factors.
[tex]\[
112 = 16 \times 7
\][/tex]
2. Rewrite the expression inside the square root:
[tex]\[
\sqrt{112 b} = \sqrt{16 \times 7 \times b}
\][/tex]
3. Simplify using the square root properties: Recall that the square root of a product is the product of the square roots:
[tex]\[
\sqrt{16 \times 7 \times b} = \sqrt{16} \times \sqrt{7} \times \sqrt{b}
\][/tex]
4. Simplify the square root of perfect squares: In this case, 16 is a perfect square:
[tex]\[
\sqrt{16} = 4
\][/tex]
So,
[tex]\[
\sqrt{16} \times \sqrt{7} \times \sqrt{b} = 4 \times \sqrt{7} \times \sqrt{b}
\][/tex]
5. Combine this result with the initial coefficient: Now, we need to multiply this by 8 from the original expression:
[tex]\[
8 \times (4 \sqrt{7} \sqrt{b})
\][/tex]
6. Multiply the constants together: Perform the multiplication:
[tex]\[
8 \times 4 = 32
\][/tex]
7. Write the final simplified form:
[tex]\[
8 \sqrt{112 b} = 32 \sqrt{7} \sqrt{b}
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
8 \sqrt{112 b} = 32 \sqrt{7} \sqrt{b}
\][/tex]
