Problem 22 Find the following products and ... [FREE SOLUTION]

91影视

Intermediate Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 5: Problem 22

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. $$ -5 \sqrt{3}(3 \sqrt{12}-9 \sqrt{8}) $$

Short Answer

Expert verified

The final expression is $-90 + 90\sqrt{6}$.

Step by step solution

Distribute the Multiplication

Distribute the multiplication of $-5\sqrt{3}$ with each term inside the parenthesis: $-5\sqrt{3}\times 3\sqrt{12} - (-5\sqrt{3}\times 9\sqrt{8})$.This results in two expressions: $-15\sqrt{3}\sqrt{12} + 45\sqrt{3}\sqrt{8}$.

Simplify Each Square Root

Now, simplify the square roots: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$ and $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$. Replace these back into the expression: $-15\sqrt{3}\times 2\sqrt{3} + 45\sqrt{3}\times 2\sqrt{2}$.

Perform the Multiplications

Now, perform the multiplications: - For the first term, $-15\sqrt{3} \times 2\sqrt{3} = -15 \times 2 \times \sqrt{3} \times \sqrt{3} = -30 \times 3 = -90$.- For the second term, $45\sqrt{3} \times 2\sqrt{2} = 45 \times 2 \times \sqrt{3}\sqrt{2} = 90\sqrt{6}$.Combine these results: $-90 + 90\sqrt{6}$.

Write the Final Expression

Combine both simplified products from step 3 to form the final expression: $-90 + 90\sqrt{6}$. This is the product expressed in the simplest radical form.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by distributing or multiplying a term across terms inside parentheses. In the exercise we are tackling here, this concept plays a crucial role in addressing the expression $-5 \sqrt{3}(3 \sqrt{12}-9 \sqrt{8})$.
Here's how it works:

The term outside the parentheses, $-5\sqrt{3}$,is applied to each term inside the parentheses, which are $3\sqrt{12}$and $9\sqrt{8}$.
This requires multiplying $-5 \sqrt{3}$each time separately by both $3\sqrt{12}$and $-9\sqrt{8}$.
This generates two new expressions: $-15\sqrt{3}\sqrt{12} + 45\sqrt{3}\sqrt{8}$.

By breaking it down this way, the distributive property allows us to handle complex multiplication in a step-by-step manner, simplifying our calculations.

Simplest Radical Form

Working with radicals often requires expressing them in their simplest form, which is crucial for clear and concise solutions. In our example problem, the radicals $\sqrt{12}$and $\sqrt{8}$need simplification.
The goal is to find the largest perfect square factor in each radical:

For $\sqrt{12}$: Decompose as $\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}$.
For $\sqrt{8}$: Decompose as $\sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2}$.

By simplifying like this, each radical is reduced to its simplest form, making them easier to work with. This method is often used in mathematics to make expressions more manageable and clearer, paving the way for the subsequent multiplication step.

Multiplying Radicals

Multiplying radicals can initially seem challenging, but it becomes manageable with proper strategy and understanding. In our specific case of simplification and multiplication, the process unfolds as follows:
First, we handle the simplified radicals obtained previously:

For $-15 \sqrt{3} \times 2 \sqrt{3}$: Multiply real numbers $-15\times 2$ and radicals $\sqrt{3} \times \sqrt{3}$ to produce $-30 \times 3 = -90$.
For $45 \sqrt{3} \times 2 \sqrt{2}$: Multiply real numbers $45 \times 2$ and radicals $\sqrt{3} \times \sqrt{2}$ to yield $90 \sqrt{6}$.

When multiplying radicals, always ensure:

The coefficients (numbers outside the radicals) are multiplied separately from the radicals themselves.
The product of like radicals results in the square of the number under the radical, simplifying further.

This approach, as illustrated in our example, ensures that expressions are always in their simplest radical form for effective and clear communication of results.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. $$ -5 \sqrt{3}(3 \sqrt{12}-9 \sqrt{8}) $$

Short Answer

Step by step solution

Distribute the Multiplication

Simplify Each Square Root

Perform the Multiplications

Write the Final Expression

Key Concepts

Distributive Property

Simplest Radical Form

Multiplying Radicals

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.