Problem 291 In the following exercises, simp... [FREE SOLUTION]

Chapter 9: Problem 291

In the following exercises, simplify. (a) \((5-\sqrt{10})^{2}\) (b) \((8+3 \sqrt{2})^{2}\)

Short Answer

Expert verified

(a) 35 - 10\sqrt{10}, (b) 82 + 48\sqrt{2}

Step by step solution

Expand the expression \( (a - b)^2 \)

Use the formula \( (a - b)^2 = a^2 - 2ab + b^2 \) on the expression \( (5 - \sqrt{10})^2 \). Here, \(a = 5\) and \(b = \sqrt{10}\).

Apply the formula

Plug values into the formula: \(5^2 - 2(5)(\sqrt{10}) + (\sqrt{10})^2 \).

Simplify each term

Calculate each term individually: \(5^2 = 25\), \(-2(5)(\sqrt{10}) = -10\sqrt{10}\), and \((\sqrt{10})^2 = 10\).

Combine terms

Combine the simplified terms to get \(25 - 10\sqrt{10} + 10 = 35 - 10\sqrt{10}\).

Expand the expression \( (a + b)^2 \)

Use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) on the expression \( (8 + 3\sqrt{2})^2 \). Here, \ a = 8 \ and \ b = 3\sqrt{2}\.

Apply the formula

Plug values into the formula: \(8^2 + 2(8)(3\sqrt{2}) + (3\sqrt{2})^2 \).

Simplify each term

Calculate each term individually: \(8^2 = 64\), \(2(8)(3\sqrt{2}) = 48\sqrt{2}\), and \((3\sqrt{2})^2 = 18\).

Combine terms

Combine the simplified terms to get \(64 + 48\sqrt{2} + 18 = 82 + 48\sqrt{2}\).

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.

square of a binomial

When dealing with the square of a binomial, it鈥檚 important to remember the formula \[ (a \pm b)^2 = a^2 \pm 2ab + b^2 \].
This formula works for both addition and subtraction.

For example, let鈥檚 consider the expression \((5 - \sqrt{10})^2\).
Here, a = 5 and b = \sqrt{10}\. You would first square each term individually, multiple the two terms, and then combine them according to the formula.

Simplifying using the formula, we get:
\[ (5 - \sqrt{10})^2 = 5^2 - 2(5)(\sqrt{10}) + (\sqrt{10})^2 \]
Which simplifies to:
\[ 25 - 10\sqrt{10} + 10 = 35 - 10\sqrt{10}\]
This gives us the final result of the expression.

expanding expressions

Expanding expressions is a fundamental skill in algebra, especially when working with binomials.
It allows you to transform a factored expression into a sum or difference that is easier to simplify.

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

In the following exercises, simplify. (a) \((5-\sqrt{10})^{2}\) (b) \((8+3 \sqrt{2})^{2}\)

Short Answer

Step by step solution

Expand the expression \( (a - b)^2 \)

Apply the formula

Simplify each term

Combine terms

Expand the expression \( (a + b)^2 \)

Apply the formula

Simplify each term

Combine terms

Key Concepts

square of a binomial

expanding expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Logic and Functions

Decision Maths

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.