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Chapter 5: Problem 7

For Problems \(1-14\), multiply and simplify where possible. $$ (-3 \sqrt{3})(-4 \sqrt{8}) $$

Short Answer

Expert verified
The expression simplifies to \(24 \sqrt{6}\).

Step by step solution

01

Simplify the Radicals

Start by simplifying each radical separately. The number inside the radical sign, known as the radicand, can often be simplified by expressing it as a product of squares.\\( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \). \Thus, the expression becomes \\(-3 \sqrt{3})(-4 \times 2 \sqrt{2}) = (-3 \sqrt{3})(-8 \sqrt{2})\).
02

Multiply the Numbers Outside the Radicals

Multiply the numerical coefficients (the numbers outside the radicals) together. \\(-3 \times -8 = 24\)
03

Multiply the Radicals Together

Multiply the radicals by applying the property \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). Here, \\(\sqrt{3} \times \sqrt{2} = \sqrt{6}\). \Thus, the expression now becomes: \\(24 \sqrt{6}\).

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Key Concepts

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

Simplifying Radicals
Simplifying radicals means breaking down the numbers inside the square root sign, or radicand, into their simplest form. This is usually achieved by expressing the radicand as a product of perfect squares and then simplifying. For example, if we have \( \sqrt{8} \), we can rewrite it as \( \sqrt{4 \times 2} \). Since 4 is a perfect square, we can further simplify it to \( \sqrt{4} \times \sqrt{2} \), which equals \( 2\sqrt{2} \).

To simplify any radical:
  • Identify perfect square factors of the radicand.
  • Rewrite the radicand as a product involving these perfect squares.
  • Take the square root of the perfect squares out of the radical.
These steps help reduce the radical to its smallest form, making further algebraic operations easier.
Radical Expressions
Radical expressions involve roots, most commonly square roots, but they can also involve cube roots, etc. For our example,聽\(-3 \sqrt{3}\) and聽\(-4 \sqrt{8}\) are both radical expressions. In these expressions, the numbers in front of the square root are coefficients, and they multiply with the roots.

Some basic properties of radical expressions include:
  • Radicals can be combined using multiplication and division, following certain algebraic properties.
  • The product rule of radicals allows multiplying under the same root, \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).
  • For different roots, sometimes expressions must first be simplified before any operations.
Understanding these properties is crucial because they dictate how we perform operations on radicals, just like regular algebraic terms.
Algebraic Multiplication Steps
When multiplying radical expressions, specific algebraic multiplication steps should be followed to ensure accuracy. Let's walk through these steps as applied in the given exercise:

  • Step 1: Simplify the Radicals - First, simplify any radicals within the expression independently, as we did with \( \sqrt{8} \) becoming \( 2\sqrt{2} \).
  • Step 2: Multiply the Coefficients - Multiply any numbers located outside the radicals. In our example, \(-3\) and \(-8\) were multiplied to get 24.
  • Step 3: Multiply the Radicals - Use the property \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \) to combine radicals. We used this property on \( \sqrt{3} \times \sqrt{2} \) to get \( \sqrt{6} \).
By following these steps, you can systematically approach and solve any problem that involves multiplying radical expressions.

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Most popular questions from this chapter

Use your calculator to estimate each of the following to the nearest one- thousandth. (a) \(7^{\frac{4}{3}}\) (b) \(10^{\frac{4}{3}}\) (c) \(12^{\frac{3}{5}}\) (d) \(19^{\frac{2}{3}}\) (e) \(7^{\frac{3}{4}}\) (f) \(10^{\frac{5}{4}}\)

Perform the indicated operations and express answers in simplest radical form. $$ \sqrt[3]{5} \sqrt{5} $$

For Problems \(33-50\), convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ (0.0037)(0.00002) $$

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

Perform the indicated operations and express answers in simplest radical form. $$ \frac{\sqrt[4]{27}}{\sqrt{3}} $$
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