Problem 263 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((-7 \sqrt{5})(-3 \sqrt{10})\) (b) \((-2 \sqrt{14})^{2}\)

Step-by-Step Solution

Answer

(a) 105 \sqrt{2}(b) 56

Step 1 - Simplify the product \( (-7 \sqrt{5})(-3 \sqrt{10}) \)

Firstly, multiply the constants: \[ (-7) \times (-3) = 21 \]Secondly, multiply the radicals: \[ \sqrt{5} \times \sqrt{10} = \sqrt{5 \times 10} = \sqrt{50} \]Combining the two results: \[ 21 \sqrt{50} \]Note that \sqrt{50} can be further simplified: \[ \sqrt{50} = \sqrt{25 \times 2} = 5 \sqrt{2} \]Thus, the simplest form is: \[ 21 \times 5 \sqrt{2} = 105 \sqrt{2} \]

Step 2 - Simplify the square \( (-2 \sqrt{14})^{2} \)

Apply the power to both the constant and the radical: \[ (-2 \sqrt{14})^{2} = (-2)^{2} \times (\sqrt{14})^{2} \]Compute the square of the constant: \[ (-2)^{2} = 4 \]And compute the square of the radical: \[ (\sqrt{14})^{2} = 14 \]Combine the two results: \[ 4 \times 14 = 56 \]

Key Concepts

Multiplying Radicals

When you multiply radicals, you follow some key steps. First, you focus on the constants (the numbers outside the radicals) and then multiply the radicals (the square roots or any other root) themselves.
For example, in the exercise part (a): \((-7 \sqrt{5})(-3 \sqrt{10})\), start by multiplying the constants \((-7) \times (-3) = 21\).
Next, work on the radicals: \(\sqrt{5} \times \sqrt{10} = \sqrt{5 \times 10} = \sqrt{50}\).
The product of \( \sqrt{50} \) can be simplified further: \( \sqrt{50} = \sqrt{25 \times 2} = 5 \sqrt{2} \). Combine this with the constants:
\21 \times 5 \sqrt{2} = 105 \sqrt{2}\.
Simplifying the process helps in tackling more complex problems efficiently.

Simplifying Square Roots

Simplifying square roots involves breaking down the number inside the root into its prime factors and finding perfect squares.
Take the example from part (a) again where \( \sqrt{50} = \sqrt{25 \times 2} \). Here, 25 is a perfect square.
You can write \( \sqrt{50} \) as \( \sqrt{25 \times 2} \) which simplifies to \( 5 \sqrt{2} \).
This technique makes it easier to work with radicals and their simplified forms.

Powers of Radicals

When dealing with radicals raised to a power, you apply the power to both the constants and the radicals separately.
For example, in the exercise part (b): \((-2 \sqrt{14})^{2}\), apply the power 2 separately:
\((-2)^{2} \) and \( (\sqrt{14})^{2} \).
Calculate these separately: \( (-2)^{2} = 4 \) and \( (\sqrt{14})^{2} = 14 \).
Then multiply the results: \4 \times 14 = 56\.
Breaking it down step by step ensures clarity and accuracy in your calculations.