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Chapter 5: Problem 89

Describe how to multiply square roots.

Short Answer

Expert verified
Two square roots can be multiplied by multiplying the radicands and then taking the square root of that product. In some cases, simplifying the square roots before multiplication can make the computation easier.

Step by step solution

01

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, will give the original number. For instance, the square root of \(9\) (written as \(\sqrt{9}\)) is \(3\), because \(3 \times 3 = 9\). Likewise, \(\sqrt{16}\) equals \(4\) because \(4 \times 4 = 16\).
02

Multiplication of Square Roots with the Same Radicand

If two square roots have the same radicand, they can be multiplied directly. For instance, \(\sqrt{5} \times \sqrt{5}\) is the same as \(\sqrt{5 \times 5}\), which simplifies to \(\sqrt{25}\), and further simplifies to \(5\).
03

Multiplication of Square Roots with Different Radicands

When the radicands are different, the square roots can still be multiplied, but the calculation might be a bit more complex. For instance, \(\sqrt{2} \times \sqrt{3}\) equals \(\sqrt{2 \times 3}\) which equals \( \sqrt{6}\). This cannot be simplified any further as \(6\) is not a perfect square.
04

Simplifying the Square Roots before Multiplication

In some cases, simplifying the square roots before multiplication can make the calculation a lot easier. This is particularly useful when dealing with larger numbers. For instance \(\sqrt{18} \times \sqrt{2}\) can be simplified and written as \(3\sqrt{2} \times \sqrt{2}\), which equals to \(3 \times \sqrt{2 \times 2}\) and further simplifies to \(3 \times \sqrt{4}\) and finally simplifies to \(3 \times 2 = 6\).

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Most popular questions from this chapter

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(0.0007,-0.007,0.07,-0.7, \ldots\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sequence \(1,4,8,13,19,26, \ldots\) is an arithmetic sequence.

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{100}\), when \(a_{1}=50, r=1\).

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{20}\), when \(a_{1}=2, r=3\).
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