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Chapter 5: Problem 1

For Problems \(1-14\), multiply and simplify where possible. $$ \sqrt{6} \sqrt{12} $$

Short Answer

Expert verified
The product is simplified to \(6\sqrt{2}\).

Step by step solution

01

Apply the Product Property of Square Roots

The product property of square roots states that for any non-negative numbers \(a\) and \(b\), \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). Here, we have \(\sqrt{6} \times \sqrt{12}\), so first multiply the values under the square roots: \(\sqrt{6 \times 12}\).
02

Perform the Multiplication Under the Square Root

Multiply the numbers under the square root: \(6 \times 12 = 72\). Therefore, \(\sqrt{6} \times \sqrt{12} = \sqrt{72}\).
03

Simplify the Square Root

To simplify \(\sqrt{72}\), start by finding the prime factorization of 72: \(72 = 2^3 \times 3^2\). Pair the factors to find the square root: \(\sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{(2^2 \times 3^2) \times 2} = 6\sqrt{2}\).

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Key Concepts

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

Product Property of Square Roots
When you have two square roots multiplied together, there's a helpful property called the Product Property of Square Roots. This property allows us to combine the roots into one. It's like a shortcut! It states: for any non-negative numbers \(a\) and \(b\), \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). So, instead of working with two separate roots, you can use this property to merge everything under a single square root sign.
  • This means multiplying the numbers inside the square roots together first.
  • For example, \(\sqrt{6} \times \sqrt{12}\) becomes \(\sqrt{6 \times 12}\).
  • This simplification gives you a new expression: \(\sqrt{72}\).
Breaking it down like this makes solving problems with square roots easier and keeps your work tidy.
Simplifying Square Roots
Simplifying square roots is all about breaking down the number inside the root until you have no more perfect squares left to factor out. Once you've combined your square roots into \(\sqrt{72}\), the next step is to simplify it. This involves looking at the number inside and finding its prime factors.
To simplify \(\sqrt{72}\), you need to find numbers that multiply together to make 72, with a focus on pairs of the same number. These pairs can be pulled out of the square root.
  • First, recognize or calculate the prime factorization: \(72 = 2^3 \times 3^2\).
  • Look for pairs in this factorization that can be converted into whole numbers. Here, you have \(2^3\) and \(3^2\).
  • Convert these pairs into their square roots: \(\sqrt{2^2} \times \sqrt{3^2} \times \sqrt{2}\).
  • This becomes \(2 \times 3 \times \sqrt{2} = 6\sqrt{2}\).
Simplifying like this gives you a cleaner answer and shows a deeper understanding of how to work with square roots.
Prime Factorization
Prime factorization is a method to express a number as the product of prime numbers. It's a fundamental step when simplifying square roots.
Let's see how it works with 72. Come up with the smallest prime number that divides 72. You might start with 2 because 72 is even:
  • 72 divided by 2 is 36.
  • 36 divided by 2 is 18.
  • 18 divided by 2 is 9.
Now, switch to the next smallest prime, 3, since 9 is not divisible by 2:
  • 9 divided by 3 is 3.
  • 3 divided by 3 is 1.
Once you reach 1, you've completed the factorization! For 72, the complete prime factorization is \(2^3 \times 3^2\). Recognizing these primes helps you with simplifying square roots by identifying pairs. It also reinforces understanding of how numbers break down, making math problems less daunting.

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Most popular questions from this chapter

For Problems \(1-18\), write each of the following in scientific notation. For example, $$ 27,800=(2.78)\left(10^{4}\right) $$ 89

Simplify each of the following. Express final results using positive exponents only. $$ \left(\frac{64 a^{\frac{1}{3}}}{16 a^{\frac{5}{9}}}\right)^{3} $$

For Problems \(1-18\), write each of the following in scientific notation. \(0.0214\)

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ \sqrt{9,000,000} $$

For Problems \(1-18\), write each of the following in scientific notation. 4290
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