Multiply: $\left(\sqrt{2 x^3}+\sqrt{12 x}\right)\left(2 \sqrt{10 x^5}+\sqrt{6 x^2}\right)$ where $x \geq 0$:
A. $2 x^2 \sqrt{5}+2 x \sqrt{3 x}+2 x^3 \sqrt{30}+3 x \sqrt{2 x}$
B. $4 x^4 \sqrt{5}+2 x^2 \sqrt{3 x}+4 x^3 \sqrt{30}+6 x \sqrt{2 x}$
C. $x^4 \sqrt{20}+x^2 \sqrt{6 x}+x^3 \sqrt{120}+x \sqrt{12 x}$
D. $2 \sqrt{10 x^4}+2 \sqrt{3 x^3}+4 \sqrt{15 x^3}+6 \sqrt{2 x}$
Answer (2)
The final result of multiplying the expression ( 2 x 3 + 12 x ) ( 2 10 x 5 + 6 x 2 ) simplifies to 4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x . The correct answer is option B.
;
Multiply the given expression: ( 2 x 3 + 12 x ) ( 2 10 x 5 + 6 x 2 ) .
Simplify each term after multiplication.
Combine the simplified terms.
The final answer is: 4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x .
Explanation
Understanding the Problem We are given the expression ( 2 x 3 + 12 x ) ( 2 10 x 5 + 6 x 2 ) to multiply, where x ≥ 0 . Our goal is to simplify the expression after performing the multiplication.
Expanding the Expression Let's multiply the two terms:
( 2 x 3 + 12 x ) ( 2 10 x 5 + 6 x 2 ) = 2 x 3 ⋅ 2 10 x 5 + 2 x 3 ⋅ 6 x 2 + 12 x ⋅ 2 10 x 5 + 12 x ⋅ 6 x 2
Simplifying Each Term Now, let's simplify each term:
Term 1: 2 x 3 ⋅ 2 10 x 5 = 2 20 x 8 = 2 4 ⋅ 5 ⋅ x 8 = 2 ⋅ 2 ⋅ x 4 ⋅ 5 = 4 x 4 5
Term 2: 2 x 3 ⋅ 6 x 2 = 12 x 5 = 4 ⋅ 3 ⋅ x 4 ⋅ x = 2 x 2 3 x
Term 3: 12 x ⋅ 2 10 x 5 = 2 120 x 6 = 2 4 ⋅ 30 ⋅ x 6 = 2 ⋅ 2 ⋅ x 3 ⋅ 30 = 4 x 3 30
Term 4: 12 x ⋅ 6 x 2 = 72 x 3 = 36 ⋅ 2 ⋅ x 2 ⋅ x = 6 x 2 x
Combining Like Terms Combining the simplified terms, we get:
4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x
Final Answer Therefore, the final simplified expression is 4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x .
Examples
Understanding how to simplify expressions with radicals is useful in many areas, such as physics and engineering, where you might need to calculate distances or areas. For example, when calculating the length of a diagonal in a rectangular prism, you often end up with radical expressions that need simplification. This skill also helps in optimizing designs and calculations in various real-world applications.
