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Magazine Chapter 5: Problem 8
Multiply and simplify where possible. \((-5 \sqrt{8})(-6 \sqrt{7})\)
Short Answer
Expert verified
The simplified product is \( 60 \sqrt{14} \).
Step by step solution
01
Simplify Inside the Radicals
First, look at the numbers inside the square roots. We have \( \sqrt{8} \) and \( \sqrt{7} \). Notice that \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \). Meanwhile, \( \sqrt{7} \) is already simplified as 7 is a prime number.
02
Rewrite the Expression
Substitute the simplified form of \( \sqrt{8} \) into the original expression: \[ (-5 \times 2\sqrt{2})(-6 \sqrt{7}) \]This simplifies to: \[ (-10\sqrt{2})(-6\sqrt{7}) \]
03
Multiply the Coefficients
Multiply the coefficients outside of the radicals. Here we have:\[ -10 \times -6 = 60 \]
04
Multiply the Radicals
Now multiply the numbers inside the square roots. We have:\[ \sqrt{2} \times \sqrt{7} = \sqrt{14} \]
05
Combine and Simplify
Combine the product of the coefficients with the product of the radicals:\[ 60 \sqrt{14} \]This is already in simplest form since 14 has no perfect square factors besides 1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Multiplying Radicals
Multiplying radicals is a step-by-step process where you handle each part of the expression carefully. Think of multiplying radicals just like you would regular numbers, but with a little extra attention to the expressions under the square root. Each radical must be treated as a whole.
For example, when multiplying \(-5 \sqrt{8}\) and \(-6 \sqrt{7}\), follow these steps:
For example, when multiplying \(-5 \sqrt{8}\) and \(-6 \sqrt{7}\), follow these steps:
- First, multiply the numbers outside the square roots (the coefficients). In our example, \(-5\) multiplied by \(-6\) gives \(30\).
- Then, multiply the numbers inside the radicals. Here, it means multiplying \(\sqrt{8}\) and \(\sqrt{7}\), which results in \(\sqrt{56}\).
Simplifying Radicals
Simplifying radicals involves reducing a radical expression to its simplest form. This is done by factoring out any perfect squares within the radical.
To simplify, look at the example \(\sqrt{8}\). Notice that \(8 = 4 \times 2\) and \(4\) is a perfect square. Therefore, \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\).
Here is a typical process to simplify any radical:
To simplify, look at the example \(\sqrt{8}\). Notice that \(8 = 4 \times 2\) and \(4\) is a perfect square. Therefore, \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\).
Here is a typical process to simplify any radical:
- Factor the number inside the radical.
- Identify any perfect squares and rewrite them outside the radical.
- Recombine to simplify.
Square Roots
Square roots are a specific type of radical, focused on finding a number that, when multiplied by itself, provides the original value under the root. Understanding square roots is essential as they frequently appear in problems related to radicals.
For instance, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\). Likewise, \(\sqrt{9} = 3\) since \(3 \times 3 = 9\). The symbol \(\sqrt{}\) indicates this relationship.
Here are some key points about square roots:
For instance, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\). Likewise, \(\sqrt{9} = 3\) since \(3 \times 3 = 9\). The symbol \(\sqrt{}\) indicates this relationship.
Here are some key points about square roots:
- Perfect squares are numbers like 1, 4, 9, 16, etc., each with whole numbers as roots.
- If a number isn't a perfect square, its square root will be an irrational number, often left as a radical.
- Recognizing perfect squares helps in simplifying radicals, which is critical when manipulating expressions.
