To simplify the expression \(\sqrt{48b^7}\), we first need to factor 48 and \(b^7\).
-
Factor 48: \[ 48 = 16 \times 3 = 4^2 \times 3 \]
-
For \(b^7\), we can break it down based on perfect squares: \[ b^7 = b^6 \times b = (b^3)^2 \times b \]
Now we can combine these factorizations: \[ 48b^7 = 16 \times 3 \times b^6 \times b = (4^2) \times 3 \times (b^3)^2 \times b \]
-
Now we can simplify the square root: \[ \sqrt{48b^7} = \sqrt{(4^2) \times 3 \times (b^3)^2 \times b} \]
-
Using the property \(\sqrt{xy} = \sqrt{x} \sqrt{y}\), we can separate the perfect squares from the square root: \[ \sqrt{48b^7} = \sqrt{(4^2)} \times \sqrt{3} \times \sqrt{(b^3)^2} \times \sqrt{b} \]
-
Now compute the square roots of the perfect squares: \[ = 4 \times \sqrt{3} \times b^3 \times \sqrt{b} \]
-
Combine the terms: \[ = 4b^3\sqrt{3b} \]
Thus, the simplified result is: \[ \boxed{4b^3\sqrt{3b}} \]
1 answer