Problem 246 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\left(\sqrt{14 x^{3}}\right)\left(\sqrt{7 x^{3}}\right)\) (b) \(\left(\sqrt{3 q^{2}}\right)\left(\sqrt{48 q^{3}}\right)\)

Step-by-Step Solution

Answer

(a) 7x^3\sqrt{2}, (b) 12q^2\sqrt{q}

Step 1 - Apply the property of square roots

Use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) to combine the square roots in each expression.For part (a): \(\left(\sqrt{14 x^{3}}\right)\left(\sqrt{7 x^{3}}\right) = \sqrt{(14 x^{3}) \cdot (7 x^{3})} \)For part (b): \(\left(\sqrt{3 q^{2}}\right)\left(\sqrt{48 q^{3}}\right) = \sqrt{(3 q^{2}) \cdot (48 q^{3})} \).

Step 2 - Multiply the expressions inside the square roots

Multiply the constants and the variables separately inside the square roots.For part (a): \(\sqrt{(14 x^{3}) \cdot (7 x^{3})} = \sqrt{98 x^{6}} \)For part (b): \(\sqrt{(3 q^{2}) \cdot (48 q^{3})} = \sqrt{144 q^{5}} \).

Step 3 - Simplify the square roots

Simplify the square roots by extracting the squares.For part (a): \(\sqrt{98 x^{6}} = \sqrt{(49 \cdot 2) x^{6}} = \sqrt{49} \cdot \sqrt{2} \cdot \sqrt{x^{6}} = 7 \cdot x^{3} \cdot \sqrt{2}=7x^{3}\sqrt{2} \)For part (b): \(\sqrt{144 q^{5}} = \sqrt{(144) q^{4} \cdot q} = \sqrt{144} \cdot \sqrt{q^{4}} \cdot \sqrt{q} = 12 \cdot q^{2} \cdot \sqrt{q} = 12q^{2}\sqrt{q} \).

Key Concepts

square roots

Square roots are one of the fundamental concepts in algebra. They answer the question 'what number multiplied by itself gives us this number?' For example, the square root of 9 is 3 because 3 multiplied by itself (3 * 3) equals 9. We denote the square root of a number 'a' as \( \sqrt{a} \). When dealing with variables, the same principle applies. For instance, \( \sqrt{x^{2}} = x \) because \( x \) times \( x \) gives \( x^{2} \).
Understanding square roots is crucial when you're asked to simplify algebraic expressions involving radicals.

multiplication of radicals

Multiplying radicals is a straightforward process once you understand square roots. If you're multiplying two square roots, you can combine them under one square root symbol using the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). For example, \( \sqrt{2}\cdot\sqrt{3} = \sqrt{2\cdot3} = \sqrt{6} \).
Applying this to our variables, combining \( \sqrt{14 x^{3}}\cdot\sqrt{7 x^{3}} \) becomes straightforward as we get \( \sqrt{(14 x^{3}) \cdot (7 x^{3})} \). The same rule applies regardless of whether the expressions inside the square roots are numbers, variables, or a combination of both.

simplification of radicals

Simplifying radicals often comes down to factoring out perfect squares. For instance, to simplify \( \sqrt{98 x^{6}} \), break it down into its component parts. We can write 98 as \( 49 \cdot 2 \) and \( x^{6} \) as \( (x^{3})^{2} \).
This gives us \( \sqrt{49 \cdot 2 \cdot (x^{3})^{2}} = \sqrt{49} \cdot \sqrt{2} \cdot \sqrt{(x^{3})^{2}} = 7 \cdot x^{3} \cdot \sqrt{2} \). Breaking it down into smaller steps like this avoids confusion and helps you handle even more complex expressions with ease.

square root properties

Several properties of square roots make simplifying expressions more manageable. Here are some crucial properties:

• \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) - This allows the multiplication of two square roots under one square root symbol.
• \( \sqrt{a^{2}} = a \) - The square root of a number squared is just the number.
• \( \sqrt{a/b} = \sqrt{a} / \sqrt{b} \) - This allows the division of two square roots to be written separately.

Using these properties, we can simplify complex expressions step-by-step. For example, to simplify \( \sqrt{144 q^{5}} \), notice that we can split \( \sqrt{144 q^{5}} \) into \( \sqrt{144} \cdot \sqrt{q^{4}} \cdot \sqrt{q} = 12 \cdot q^{2} \cdot \sqrt{q} \). Leveraging these properties ensures our expressions are fully simplified and more manageable.