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Chapter 2: Problem 135

Show that \(\sqrt{9-4 \sqrt{5}}=\sqrt{5}-2\)

Short Answer

Expert verified
To show that \(\sqrt{9-4 \sqrt{5}}=\sqrt{5}-2\), we square both sides and simplify. \[\left(\sqrt{9-4 \sqrt{5}}\right)^2= \left(\sqrt{5}-2\right)^2\] Expanding the right side using the formula (a - b)^2 = a^2 - 2ab + b^2, where a = \(\sqrt{5}\) and b = 2, we get: \[5 - 4\sqrt{5} + 4 = 9 - 4\sqrt{5}\] Comparing both sides, we see that they are equal, which confirms the given equation: \[\sqrt{9-4 \sqrt{5}}=\sqrt{5}-2\]

Step by step solution

01

Square both sides of the equation

Squaring both sides will eliminate the square root on the left side, and allow us to process with further simplification. \[\left(\sqrt{9-4 \sqrt{5}}\right)^2= \left(\sqrt{5}-2\right)^2\]
02

Simplify the right side of the equation

We will expand the right side of the equation using the formula (a - b)^2 = a^2 - 2ab + b^2, where a = \(\sqrt{5}\) and b = 2. \[\left(\sqrt{5}-2\right)^2 = (\sqrt{5})^2 - 2(\sqrt{5})(2) + (2)^2\] Then, simplify: \[= 5 - 4\sqrt{5} + 4\]
03

Combine like terms on the right side

Combine the constants: \[= 9 - 4\sqrt{5}\]
04

Compare both sides of the equation

Now, compare the simplified expressions on both sides. The left side is \(9 - 4\sqrt{5}\), and the right side is \(9 - 4\sqrt{5}\). Since both sides are equal, this demonstrates that: \[\sqrt{9-4 \sqrt{5}}=\sqrt{5}-2\]

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Key Concepts

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

Square Roots
Square roots are a fundamental concept in mathematics, especially in algebra. They represent a number that, when multiplied by itself, gives the original number under the square root symbol. For example, the square root of 16 is 4 because 4 times 4 is 16.
It's like working backwards from squaring a number. In our exercise, \(\sqrt{9-4 \sqrt{5}}\) is the part getting squared to eliminate the square root; this is because squaring and square roots are inverse operations.
  • Square roots can be both perfect, like \(\sqrt{16} = 4\), and imperfect, like \(\sqrt{5}\).
  • They are often used in equations where the solution involves finding a value that when squared gives the original value.
Recognizing the operation between square roots and squaring provides a good foundation for tackling algebraic expressions effectively.
Simplification
Simplification is a process used in mathematics to make expressions easier to work with. It's like cleaning up an algebraic mess to make solving problems simpler. In our case, we've performed simplification when we worked through the equation \(\left(\sqrt{5}-2\right)^2\) in the original exercise.
The trick is to break down expressions using known algebraic identities such as \( (a-b)^2 = a^2 - 2ab + b^2 \). Applying this lets us write simpler, equivalent expressions.
  • Using formulas and identities correctly is a key aspect of simplification.
  • It's important to remember that merging like terms (terms with the same variables and exponents) is a big part of this process.
In this exercise, simplification boiled down complex terms into directly comparable elements, leading to clearer equations.
Equations
An equation is a mathematical statement asserting the equality of two expressions. They are at the core of algebra. Solving equations means finding the value of variables that make the equation true. In our example, by showing \(\sqrt{9-4 \sqrt{5}}=\sqrt{5}-2\) is true, we proved that both expressions were equal.
Using equations involves operations such as addition, squaring, multiplication, etc., to relate different expressions.
  • Equations are often used to solve problems or verify identities. The goal is to manipulate them using valid operations to simplify or prove a statement.
  • Key operations, like squaring both sides, are often used to handle complex elements, such as square roots, efficiently.
Working with equations requires careful attention to detail in each operation to ensure valid transformations lead to true statements.
Square of a Binomial
The square of a binomial is a specific algebraic form that occurs frequently in mathematics. It's given by the formula \( (a-b)^2 = a^2 - 2ab + b^2 \), where you'll expand a binomial (two-term expression) squared. This formula helps simplify expressions and solve equations like the one in our exercise.
When we dealt with \(\left(\sqrt{5}-2\right)^2\), we used this formula to transform an expression into a sum of terms that is easier to handle.
  • The square of a binomial expands into three distinct parts - the square of the first term, twice the product of the two terms, and the square of the second term.
  • It's important to remember this identity because it allows for rewriting expressions in a straightforward form that is much easier to manage.
Understanding the square of a binomial provides powerful tools for tackling more involved algebraic problems by breaking them down into simpler, solvable components.

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Most popular questions from this chapter

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\) $$ f(x)=\frac{3}{x^{2}} $$

Evaluate the indicated quantities. Do not use a calculator-pushing buttons for these exercises will not help you understand rational powers. $$ 32^{-4 / 5} $$

Evaluate the indicated quantities. Do not use a calculator-pushing buttons for these exercises will not help you understand rational powers. $$ 81^{3 / 4} $$

Suppose \(m\) is an odd integer. Show that the function \(f\) defined by \(f(x)=x^{m}\) is an odd function.

Expand the expression. $$ (5+\sqrt{x})^{2} $$
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