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

Expert–verified

Solution:

Step 1

Express $32 b^{2}$ as $(4 b)^{2} \cdot 2$ .

Step 1.1

Extract the square of $16$ from $32$ . $4 \sqrt{16 \cdot 2 b^{2}}$

Step 1.2

Represent $16$ as $4^{2}$ . $4 \sqrt{4^{2} \cdot 2 b^{2}}$

Step 1.3

Rearrange the terms. $4 \sqrt{4^{2} b^{2} \cdot 2}$

Step 1.4

Express $4^{2} b^{2}$ as $(4 b)^{2}$ . $4 \sqrt{(4 b)^{2} \cdot 2}$

Step 2

Extract terms from under the radical sign. $4 (4 b \sqrt{2})$

Step 3

Multiply $4$ by $4 b$ . $16 b \sqrt{2}$

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 $4 \sqrt{32 b^{2}}$ 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.