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Chapter 6: Problem 22

Combine radicals, if possible. \(-6 \sqrt{98}+4 \sqrt{8}\)

Short Answer

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Question: Simplify and combine the given expression: \(-6 \sqrt{98} + 4 \sqrt{8}\). Answer: \(-34 \sqrt{2}\).

Step by step solution

01

Simplify each radical

First, let's simplify each radical term separately. We can rewrite each expression by factoring out the largest perfect square from inside the square root. For the first term, \(-6 \sqrt{98}\): \(-6 \sqrt{98} = -6 \sqrt{2 \cdot 49}\), and since \(49\) is a perfect square, we can rewrite the expression as: \(-6 \sqrt{2 \cdot 49} = -6 \sqrt{2} \cdot \sqrt{49} = -6 \sqrt{2} \cdot 7\) For the second term, \(4 \sqrt{8}\): \(4 \sqrt{8} = 4 \sqrt{2 \cdot 4}\), and since \(4\) is a perfect square, we can rewrite the expression as: \(4 \sqrt{2 \cdot 4} = 4 \sqrt{2} \cdot \sqrt{4} = 4 \sqrt{2} \cdot 2\)
02

Combine simplified radicals

Now that we have simplified each term, let's combine them: \((-6 \sqrt{2} \cdot 7) + (4 \sqrt{2} \cdot 2) = -42 \sqrt{2} + 8 \sqrt{2}\) Since both terms contain the \(\sqrt{2}\), we can combine them by adding their coefficients: \((-42 \sqrt{2}) + (8 \sqrt{2}) = (-42 + 8) \sqrt{2} = -34 \sqrt{2}\) So, the combined expression is: \(-6 \sqrt{98} + 4 \sqrt{8} = -34 \sqrt{2}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radical Expressions
Radical expressions involve root values, predominantly square roots, cube roots, etc. Simply put, radicals are expressions that contain a symbol \sqrt{ } to denote a root. Working with radical expressions often requires simplifying them for easier calculations.
  • Consider \(-6 \sqrt{98}\). Here, \sqrt{98} is a radical expression.
  • It can be simplified by factoring out any perfect squares, which are numbers like 1, 4, 9, 16, etc., whose square root is also a whole number.
  • For \sqrt{98}, since 98 can be expressed as \2 \times 49\, and 49 is a perfect square, it simplifies to \7 \sqrt{2}\.
Simplifying radicals allows for more straightforward calculations, especially when combining terms or solving equations.
Perfect Squares
Perfect squares are critical in simplifying radical expressions. A perfect square is a number that is the square of an integer. For example:
  • 4 is a perfect square since it equals \(2^2\).
  • 36 is another example, as \(6^2 = 36\).
In simplifying radicals, recognizing numbers that are perfect squares helps to break them down. In our problem, \(-6 \sqrt{98} = -6 \sqrt{2 \cdot 49}\) because 49 is a perfect square that equates to \7^2\. Utilizing perfect squares makes it possible to rewrite radical expressions more simply, helping to convert conjugated multiplication into simpler terms, easing the process of combining like terms.
Combining Like Terms
Combining like terms is fundamental when simplifying and streamlining expressions, including those involving radicals. Like terms are terms that contain the same variable raised to the same power.
  • For radicals, like terms share the same radicand (the number inside the radical).
  • In our example, both \(-42 \sqrt{2}\) and \(8 \sqrt{2}\) have the radicand \sqrt{2}\. Thus, they can be combined by adding their coefficients: \(-42 + 8 = -34\).
Thus, the expression \(-6 \sqrt{98} + 4 \sqrt{8}\) ultimately simplifies to \(-34 \sqrt{2}\). Combining like terms is crucial for cleaning up mathematical expressions, making them easier to interpret and solve.

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