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Chapter 1: Problem 3

Express each in terms of the simplest possible radical. $$\sqrt{18} \times \sqrt{10}$$

Short Answer

Expert verified
The simplest radical form is \( 6\sqrt{5} \).

Step by step solution

01

Multiply the radicands

To simplify the expression \( \sqrt{18} \times \sqrt{10} \), we first multiply the numbers inside the square roots. The result of multiplying the radicands 18 and 10 is 180. So, we have: \( \sqrt{18} \times \sqrt{10} = \sqrt{180} \).
02

Simplify the square root

Now, simplify \( \sqrt{180} \). First, we factor 180 into its prime factors: \( 180 = 2^2 \times 3^2 \times 5 \). Taking the square root of these factors allows us to remove the perfect squares: \( \sqrt{2^2} = 2 \) and \( \sqrt{3^2} = 3 \). Therefore, we have \( \sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5} \).

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Key Concepts

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

Multiplying Radicals
When you see a mathematical expression that involves radicals, such as \( \sqrt{a} \times \sqrt{b} \), the first step is often to multiply the radicands. The radicands are the numbers found inside the square root symbols. In our example, \( a = 18 \) and \( b = 10 \), so you multiply these to get \( 180 \).
This means \( \sqrt{18} \times \sqrt{10} = \sqrt{180} \). By multiplying the radicands, you have effectively transformed a product of radicals into a single simplified radical expression, making the next steps in your calculations more straightforward.
  • This technique works not just for square roots, but for radicals of any index.
  • Simplifying early can save time and effort on more complex problems.
Remember, the multiplication of radicands under the same type of radical is always the same: just multiply the numbers inside!
Prime Factorization
After multiplying the radicands to obtain \( 180 \), the next important move is prime factorization. This process involves breaking down a number into the smallest prime numbers that can be multiplied together to give the original number.
For \( 180 \), we can express it as a product of primes: \( 180 = 2^2 \times 3^2 \times 5 \).

Prime factorization is crucial because:
  • It allows us to identify and pull out perfect squares, making the square root easier to simplify.
  • It reveals the building blocks of any given number, useful for various kinds of calculations.
Through prime factorization, you gain a clear path to further breaking down radicals.
Square Roots
Once you have the prime factorization in hand, the final step to simplify a radical like \( \sqrt{180} \) is to take out the square roots of the perfect square factors. In this case, \( 180 = 2^2 \times 3^2 \times 5 \).
The perfect squares here are \( 2^2 \) and \( 3^2 \). When you take the square root of \( 2^2 \), you get \( 2 \), and \( \sqrt{3^2} \) gives \( 3 \). This leaves us with a simplified form of:
\[ \sqrt{180} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5}. \]
It's worth noting:
  • Only perfect squares inside the radicand can be taken out as a whole number.
  • The remaining non-square factor, in this case \( \sqrt{5} \), stays under the radical symbol.
Understanding this concept ensures that you can simplify any square root efficiently.

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Most popular questions from this chapter

Determine whether each statement is true for all real numbers \(x\). If the statement is false, then indicate one counterexample, i.e. a value of \(x\) for which the statement is false. $$-x \leq 0$$

Use scientific notation and the laws of exponents to perform the indicated operations. Give the result in scientific notation rounded to two significant figures. $$\frac{\left(1 \times 10^{-3}\right)\left(3.28 \times 10^{6}\right)}{4 \times 10^{7}}$$

Perform the indicated operation and simplify. $$\frac{8}{9-x^{2}} \div \frac{2 x}{x^{3}-x^{2}-6 x}$$

Find all values of \(x\) that make the equation true. $$|6-x|=10$$

Perform the indicated operation and simplify. $$\frac{1}{(x-3)^{2}}-\frac{3}{x-3}$$
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