Problem 202 - Elementary Algebra

Question

In the following exercises, simplify. $$ 8 \sqrt{72 q^{6}}-3 \sqrt{75 q^{6}} $$

Step-by-Step Solution

Answer

48 \sqrt{2} q^3 - 15 \sqrt{3} q^3

Step 1 – Factor Inside the Radicals

Start by factoring the numbers inside each square root to their prime factors, and rewrite the expression as: \(8 \sqrt{72q^{6}} - 3 \sqrt{75q^{6}}\) We know that 72 can be factored into 36 and 2 (since 36 is a perfect square) and 75 can be factored into 25 and 3 (since 25 is a perfect square).

Step 2 – Rewrite Using Perfect Squares

Rewrite each square root by separating out the perfect squares: \(8 \sqrt{36 \times 2 \times q^6} - 3 \sqrt{25 \times 3 \times q^6}\)

Step 3 – Simplify the Radicals

Simplify the square roots by taking the square root of the perfect square factors: \(8 \times 6 \times \sqrt{2} \times q^3 - 3 \times 5 \times \sqrt{3} \times q^3\) This simplifies to: \(48 \sqrt{2} q^3 - 15 \sqrt{3} q^3\)

Step 4 – Combine Like Terms

Notice that there are no like terms to combine (because the radicals have different values inside), so the simplified expression is as follows: \(48 \sqrt{2} q^3 - 15 \sqrt{3} q^3\)

Key Concepts

factoring under a radical

Factoring under a radical means breaking down the number inside the radical (square root in this case) into its prime factors. This helps in simplifying the expression. For instance, in the given exercise, we had the terms \(\sqrt{72}\) and \(\sqrt{75}\).

To factor 72, we recognized it can be broken down into 36 (which is a perfect square) and 2: \(\sqrt{72} = \sqrt{36 \times 2}\).
Similarly, for 75, it can be broken down into 25 (another perfect square) and 3: \(\sqrt{75} = \sqrt{25 \times 3}\).
The equation then becomes \(8 \sqrt{36 \times 2 \times q^6} - 3 \sqrt{25 \times 3 \times q^6}\).

Factoring aids in simplifying the radicals more effectively, which brings us to the next step.

simplifying square roots

Simplifying square roots involves extracting the square roots of perfect square factors from within the radical. The process makes the expression easier to manage.
In our example, once we've factored 72 and 75, we proceed to extract their square roots. Here’s how we do it: \(8 \sqrt{36 \times 2 \times q^6} - 3 \sqrt{25 \times 3 \times q^6}\).
\(8 \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{q^6} - 3 \sqrt{25} \cdot \sqrt{3} \cdot \sqrt{q^6}\).

We know the square root of 36 is 6, the square root of 25 is 5, and the square root of \(q^6\) is \(q^3\) (since \(\sqrt{q^6} = q^3\)).

Thus, \(8 \times 6 \times \sqrt{2} \times q^3 - 3 \times 5 \times \sqrt{3} \times q^3\),
which simplifies further to \(48 \sqrt{2} q^3 - 15 \sqrt{3} q^3\).

Now that we have simplified the square roots, we can move to the final concept – combining like terms.

combining like terms

To combine like terms, the terms must have the same radical and variable parts. In the expression \(48 \sqrt{2} q^3 - 15 \sqrt{3} q^3\), the two terms do not have the same radical components (the radicals are \(\sqrt{2}\) and \(\sqrt{3}\), which are different).

Thus, these terms cannot be combined.
Having different radical parts makes the terms 'unlike' terms, meaning we leave them as they are.
Therefore, the final simplified expression is: \(48 \sqrt{2} q^3 - 15 \sqrt{3} q^3\).

Understanding when and when not to combine terms is crucial in simplifying expressions efficiently. This ensures clarity and accuracy in your solutions.