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Chapter 7: Problem 56

Simplify the products. Give exact answers. $$(\sqrt{2}+5)(\sqrt{2}+5)$$

Short Answer

Expert verified
The simplified form of \( (\sqrt{2} + 5)(\sqrt{2} + 5) \) is \ 27 + 10 \sqrt{2}.\

Step by step solution

01

Expand the Expression

First, recognize that \( \sqrt{2} + 5 \) is a binomial. Use the distributive property or the formula for the square of a binomial: \[ (a + b)^2 = a^2 + 2ab + b^2 \] Here, \( a = \sqrt{2} \) and \( b = 5 \).
02

Calculate Each Term Individually

Calculate each term in the expansion: \[ a^2 = (\sqrt{2})^2 = 2 \] \[ 2ab = 2 \times \sqrt{2} \times 5 = 10 \sqrt{2} \] \[ b^2 = 5^2 = 25 \ \]
03

Combine the Terms

Now, add all the terms together: \[ a^2 + 2ab + b^2 = 2 + 10 \sqrt{2} + 25 \] Combine the like terms to get the final answer.
04

Simplify the Expression

Combine the constants and the radical: \[ 2 + 25 + 10 \sqrt{2} = 27 + 10 \sqrt{2} \] This is the simplified form of the expression.

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Key Concepts

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

Binomial Expansion
Binomial expansion involves expanding expressions that are raised to a power. In our original problem, we are expanding the binomial \( ( \sqrt{2} + 5 ) \) squared. When a binomial is squared, we can use the special formula: \[ ( a + b )^2 = a^2 + 2ab + b^2 \] Here, \(a = \sqrt{2} \) and \(b = 5 \).
You substitute these values into the formula to expand the expression completely. This method helps in breaking down and simplifying the expression step-by-step.
Distributive Property
The distributive property allows you to multiply each term inside a parenthesis by another term outside. It's essential for expanding binomials and simplifying expressions.
In our example, to simplify \( ( \sqrt{2} + 5 )( \sqrt{2} + 5 ) \), you distribute each term in the first binomial to every term in the second binomial. This process ensures that all parts are properly expanded and combined:
  • First, multiply \( \sqrt{2} \) by \( \sqrt{2} \) to get \[ ( \sqrt{2} )^2 \]
  • Next, distribute \( 5 \) to \( \sqrt{2} \) and vice-versa for the middle terms, resulting in \[ 2 \times \sqrt{2} \times 5 \]
  • Finally, multiply the constants \( (5 \times 5) \) to get the last term \[ b^2 \]
This property is core to performing algebraic operations efficiently.
Radical Expressions
Radical expressions include the root symbol, often the square root. In our problem, \( \sqrt{2} \) is a radical expression. Simplifying these requires care:
  • Square \( \sqrt{2} \): \[ ( \sqrt{2} )^2 = 2 \]
  • Handle like terms in radical form: combining them carefully (e.g., \(10 \sqrt{2} \) remains as is)
This ensures radicals are managed correctly and do not combine incorrectly with non-radical terms. Remember: radicals with different radicands (numbers under the root symbol) cannot be directly combined.
Algebraic Operations
Algebraic operations like addition, subtraction, multiplication, and division involve symbols and letters representing numbers. In our example, several algebraic operations simplify the binomial expression:
  • Squaring terms: involves multiplying the term by itself.
  • Multiplying constants and radicals: \2 \times \sqrt{2} \times 5 \ gives \ 10 \sqrt{2}
  • Combining like terms: \[ 2 + 25 = 27 \]

Using proper rule application and organization ensures the expression simplifies smoothly. These principles apply universally across algebra to handle mathematical expressions.

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Most popular questions from this chapter

Simplify. $$\frac{\sqrt{h k}}{\sqrt{h}+3 \sqrt{k}}$$

Solve each equation. $$(x-3)^{2 / 3}=-4$$

Top stock fund. The average annual return \(r\) is a function of the initial investment \(P,\) the number of years \(n,\) and the amount \(S\) that it is worth after \(n\) years: $$r=\left(\frac{S}{P}\right)^{1 / n}-1$$ An investment of \(\$ 10,000\) in the World Precious Minerals Fund in 2001 was worth \(\$ 62,760\) in 2006 (www.money.com). Find the 5 -year average annual return.

Because 3 is the square of \(\sqrt{3}\), a binomial such as \(y^{2}-3\) is a difference of two squares. a) Factor \(y^{2}-3\) and \(2 a^{2}-7\) using radicals. b) Use factoring with radicals to solve the equation \(x^{2}-8=0\). c) Assuming \(a\) is a positive real number, solve the equation \(x^{2}-a=0\).

Solve each equation. $$(t-1)^{-2 / 3}=2$$
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