Simplify 4 square root of 32b^2
The problem provided is asking you to perform a mathematical simplification on the expression presented. In the expression "4 square root of 32b^2", you're expected to simplify both the numerical part under the square root and the variable part. This involves finding the primary square factor of 32 and simplifying it along with the 4 outside the square root, and also simplifying the square root of the variable part "b^2". The expectation is to rewrite the expression in its simplest form, expressing any perfect squares as their square root and combining them with the coefficient outside the square root.
$4 \sqrt{32 b^{2}}$
Answer
Solution:
Step 1
Express $32b^2$ as $(4b)^2 \cdot 2$.
Step 1.1
Extract the square of $16$ from $32$. $4 \sqrt{16 \cdot 2b^2}$
Step 1.2
Represent $16$ as $4^2$. $4 \sqrt{4^2 \cdot 2b^2}$
Step 1.3
Rearrange the terms. $4 \sqrt{4^2b^2 \cdot 2}$
Step 1.4
Express $4^2b^2$ as $(4b)^2$. $4 \sqrt{(4b)^2 \cdot 2}$
Step 2
Extract terms from under the radical sign. $4(4b\sqrt{2})$
Step 3
Multiply $4$ by $4b$. $16b\sqrt{2}$
Knowledge Notes:
To simplify an expression involving square roots, we can use the following knowledge points:
Factorization: Breaking down a number into its prime factors can help simplify the square root. For instance, $32$ can be factored into $16 \times 2$, where $16$ is a perfect square.
Perfect Squares: Recognizing perfect squares, such as $16 = 4^2$, is crucial because the square root of a perfect square is an integer.
Properties of Square Roots: The square root of a product is equal to the product of the square roots of the individual factors, provided that all the quantities under the square root are non-negative.
Simplifying Radicals: When a term under a radical can be expressed as a square of another term, it can be taken out of the radical, simplifying the expression.
Combining Like Terms: After simplifying the radical, we can combine like terms outside the radical to further simplify the expression.
Algebraic Manipulation: Multiplying terms outside the radical to get the final simplified form of the expression.
In this problem, we applied these principles to simplify the expression $4\sqrt{32b^2}$ by factoring out the perfect square from under the radical and then simplifying the expression by pulling terms out from under the radical and multiplying them together.
