Problem 20 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$\sqrt{98 x^{2}}-\sqrt{72 x^{2}}$$

Step-by-Step Solution

Answer

The simplified form is \(x\sqrt{2}\).

Step 1: Factor the Square Roots

We need to express each term under the square root as a product of squares. For \(\sqrt{98x^2}\), we have \(98 = 2 \times 49\), so \(\sqrt{98x^2} = \sqrt{2 \times 49 \times x^2}\). For \(\sqrt{72x^2}\), we have \(72 = 2 \times 36\), so \(\sqrt{72x^2} = \sqrt{2 \times 36 \times x^2}\).

Step 2: Simplify the Square Roots

Simplify each square root by taking the square out of the radical. For \(\sqrt{98x^2}\), this becomes \(\sqrt{49} \times \sqrt{2} \times \sqrt{x^2} = 7x\sqrt{2}\). For \(\sqrt{72x^2}\), this becomes \(\sqrt{36} \times \sqrt{2} \times \sqrt{x^2} = 6x\sqrt{2}\).

Step 3: Combine the Terms

We now have the expression \(7x\sqrt{2} - 6x\sqrt{2}\). Since both terms have \(x\sqrt{2}\) in common, we can combine them: \((7x - 6x)\sqrt{2} = x\sqrt{2}\).

Key Concepts

Factoring

Factoring is a great tool in algebra that allows us to break down complex terms into simpler components. When dealing with square roots, factoring helps by identifying perfect squares that simplify the radical expression.
To factor a number, break it down into its prime number components and find if there are any perfect squares. For example, in the given problem, the number 98 can be factored as \( 2 \times 49 \), where 49 is a perfect square. Similarly, 72 can be factored into \( 2 \times 36 \), where 36 is a perfect square.
Whenever you encounter a square root, it's helpful to factor the expression inside the square root first. Finding these squares enables you to simplify further by taking the square root of the perfect square terms.

Combining Like Terms

Combining like terms in algebra is a handy method to simplify expressions further. It involves merging terms with the same variables raised to the same power.
In radical expressions, like terms share the same non-radical and radical parts. In our problem, after simplifying, both expressions turned into \(7x\sqrt{2}\) and \(6x\sqrt{2}\). These terms are like terms because they share the \(x\sqrt{2}\) part.
Once identified, like terms can be added or subtracted. This means you treat the coefficients (the numbers in front) like normal arithmetic. Here, you subtract the similar terms:

• Subtract \(6x\sqrt{2}\) from \(7x\sqrt{2}\), which results in \( (7x - 6x)\sqrt{2} \)
• The result becomes \(x\sqrt{2}\)

Combining like terms helps in streamlining expressions into a simpler form, which is easier to understand and work with.

Radical Expressions

Radical expressions involve roots, such as square roots, and they can initially look complex. However, once you understand the basics of simplifying them, they become far more manageable. The key is breaking the expression down into simpler parts and finding ways to deal with the radicals.
In the problem \(\sqrt{98x^2} - \sqrt{72x^2}\), we first simplify the square roots. To handle this, we factor the numbers under the radicals into their perfect square components.

• The square root of a perfect square is straightforward; for instance, the square root of 49 is 7.
• Also, the square root of \(x^2\) is \(x\).

Now, combine and simplify:

• For \(\sqrt{98x^2}\), it simplifies to \(7x\sqrt{2}\).
• For \(\sqrt{72x^2}\), it simplifies to \(6x\sqrt{2}\).

Express these terms together. This simplification uses the property of radicals and brings your expression into a more user-friendly form. Always remember that radical expressions can often be reduced significantly with diligent factoring and identification of like terms.