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Chapter 4: Problem 123

(a)(i) For any positive real numbers \(a, b,\) prove that $$ \sqrt{a}+\sqrt{b}=\sqrt{a+b+\sqrt{4 a b}} $$ (ii) Simplify \(\sqrt{5+\sqrt{24}}\). (b) (i) Find a similar formula for \(\sqrt{a}-\sqrt{b}\). (ii) Simplify \(\sqrt{5-\sqrt{16}}\) and \(\sqrt{6-\sqrt{20}}\).

Short Answer

Expert verified
(a)(i) Proven as true. (ii) \( \sqrt{3} + \sqrt{2} \). (b)(i) \( \sqrt{a} - \sqrt{b} = \sqrt{a+b-\sqrt{4ab}} \). (ii) \(1, \sqrt{5}-1\).

Step by step solution

01

Understanding the equation

We need to prove that \( \sqrt{a} + \sqrt{b} = \sqrt{a+b+\sqrt{4ab}} \). Let's start by squaring both sides.
02

Squaring the left side and right side

First, square the left side: \( (\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + b \). Now, square the right side: \( (\sqrt{a+b+\sqrt{4ab}})^2 = a+b+\sqrt{4ab} \).
03

Equating the squared expressions

Both squared expressions result in \( a + 2\sqrt{ab} + b \) and \( a + b + \sqrt{4ab} \). Notice that \( 2\sqrt{ab} = \sqrt{4ab} \). Thus, both sides are indeed equal.
04

Proving both sides are equal

Since the squared expressions are equal, \( \sqrt{a} + \sqrt{b} = \sqrt{a+b+\sqrt{4ab}} \) holds true.
05

Simplifying \( \sqrt{5+\sqrt{24}} \)

Assume \( \sqrt{x} + \sqrt{y} = \sqrt{5 + \sqrt{24}} \). By comparing with the proved formula, set \( x = 3 \) and \( y = 2 \), since \( \sqrt{4xy} = \sqrt{24} \) fits as \( \sqrt{12} = 2*\sqrt{xy} \), giving \( \sqrt{5+\sqrt{24}} = \sqrt{3} + \sqrt{2} \).
06

Finding a similar formula for \( \sqrt{a}-\sqrt{b} \)

By squaring \( \sqrt{a}-\sqrt{b} \), we find: \( (\sqrt{a} - \sqrt{b})^2 = a + b - 2\sqrt{ab} \), which implies \( \sqrt{a}-\sqrt{b} = \sqrt{a+b-\sqrt{4ab}} \).
07

Simplifying \( \sqrt{5-\sqrt{16}} \)

Assume \( \sqrt{x} - \sqrt{y} = \sqrt{5-\sqrt{16}} \). Setting \( x = 4 \) and \( y = 1 \), we achieve \( \sqrt{5-\sqrt{16}} = \sqrt{4} - \sqrt{1} = 2 - 1 = 1 \).
08

Simplifying \( \sqrt{6-\sqrt{20}} \)

Assume \( \sqrt{x} - \sqrt{y} = \sqrt{6-\sqrt{20}} \). Choosing \( x = 5 \) and \( y = 1 \), we confirm \( \sqrt{6-\sqrt{20}} = \sqrt{5} - \sqrt{1} \), resulting in \( \sqrt{5} - 1 \).

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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 fundamental components in algebra and serve as the inverse operation to squaring a number. Essentially, a square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, since \(3 \times 3 = 9\). When working with square roots:
  • It's important to understand that every positive real number has both a positive and a negative square root. Although, in many mathematical contexts, we only consider the principal (positive) square root.
  • Square roots are involved in various mathematical expressions and equations, making them crucial in solving different problems.
  • The notation \( \sqrt{a} \) denotes the square root of the number \(a\).
The given equation \(\sqrt{a} + \sqrt{b} = \sqrt{a+b+\sqrt{4ab}}\) requires a solid understanding of square roots and their properties to solve. This includes knowing how to manipulate and equate expressions involving square roots.
Simplifying Radical Expressions
Simplifying radical expressions often involves breaking down the radical into its simplest form. This is an essential skill in algebra because it helps in making complex expressions more manageable. To understand the process, consider the expression from the exercise: \(\sqrt{5 + \sqrt{24}}\). Simplifying this involves identifying possible perfect squares and using them to break down the expression.Key steps in simplifying radical expressions include:
  • Looking for ways to express the radicand (the number under the root) in terms of perfect squares.
  • Using properties of square roots, such as \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), to simplify the expression efficiently.
  • Testing different values to ensure they fit the derived formulas鈥攆or instance, assuming \(\sqrt{x} + \sqrt{y}\) for the given expression and comparing terms.
Understanding these methods can help simplify complex radical expressions like \(\sqrt{5+\sqrt{24}} = \sqrt{3} + \sqrt{2}\), leading to clearer and more concise solutions.
Proving Equalities
Proving equalities in algebra involves demonstrating that two expressions are equivalent, usually through algebraic manipulation. This often requires squaring both sides, distributing terms, and simplifying resulting expressions, as shown in part (a) of the exercise.Here鈥檚 how to approach proving an equality:
  • Identify what each side of the equation represents and what needs to be shown as equal.
  • Use algebraic rules, such as squaring binomials and distributing terms, to manipulate the expressions. For example, the exercise involved proving \(\sqrt{a} + \sqrt{b} = \sqrt{a+b+\sqrt{4ab}}\) by squaring both sides.
  • Compare the resulting expressions after manipulation to see if they match. This involves checking whether distributed terms simplify into identical expressions on both sides of the equation.
The solution involves recognizing that \(2\sqrt{ab}\) is indeed \(\sqrt{4ab}\), thereby confirming the given expression's validity. These techniques are foundational in algebra and are crucial for validating the relationships between expressions in various mathematical scenarios.

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Most popular questions from this chapter

The equation \(x^{3}+3 x^{2}-4=0\) clearly has " \(x=1\) " as a positive solution. (The other two solutions are \(x=-2,\) and \(x=-2-\) a repeated root; however negatives were viewed with suspicion in the sixteenth century, so this root might well have been ignored.) Try to understand how the following sequence of moves "finds the root \(x=1 "\) : (i) substitute \(y=x+1\) to get a cubic equation in \(y\) with no term in \(y^{2}\); (ii) imagine \(y=u+v\) and interpret the identity for $$ (u+v)^{3}=u^{3}+3 u v(u+v)+v^{3} $$ as your cubic equation in \(y\); (iii) solve the simultaneous equations " \(3 u v=3 ", " u^{3}+v^{3}=2\) " (not by guessing, but by substituting \(v=\frac{1}{u}\) from the first equation into the second to get a quadratic equation in " \(u^{3}\) ", which you can then solve for \(u^{3}\) before taking cube roots); (iv) then find the corresponding value of \(v\), hence the value of \(y=u+v,\) and hence the value of \(x\). The simple method underlying Problem 135 is in fact completely general. Given any cubic equation $$ a x^{3}+b x^{2}+c x+d=0 \quad \text { (with } \left.a \neq 0\right) $$ we can divide through by \(a\) to reduce this to $$ x^{3}+p x^{2}+q x+r=0 $$ with leading coefficient \(=1 .\) Then we can substitute \(y=x+\frac{p}{3}\) and reduce this to a cubic equation in \(y\) $$ y^{3}-3\left(\frac{p}{3}\right)^{2} y+q y+\left[r+2\left(\frac{p}{3}\right)^{3}-q\left(\frac{p}{3}\right)\right]=0 $$ which we can treat as having the form $$ y^{3}-m y-n=0 $$ So we can set \(y=u+v\) (for some unknown \(u\) and \(v\) yet to be chosen), and treat the last equation as an instance of the identity $$ (u+v)^{3}-3 u v(u+v)-\left(u^{3}+v^{3}\right)=0 $$ which it will become if we simply choose \(u\) and \(v\) to solve the simultaneous equations $$ 3 u v=m, \quad u^{3}+v^{3}=n . $$ We can then solve these equations to find \(u,\) then \(v-\) and hence find \(y=u+v\) and \(x=y-\frac{p}{3}\).

(a) Mark on the coordinate line all those points \(x\) in the interval [0,1) which have the digit "1" immediately after the decimal point in their decimal expansion. What fraction of the interval [0,1) have you marked? Note: " [0,1) " denotes the set of all points between 0 and 1 , together with \(0,\) but not including 1. [0,1] denotes the interval including both endpoints; and (0,1) denotes the interval excluding both endpoints. (b) Mark on the interval [0,1) all those points \(x\) which have the digit "1" in at least one decimal place. What fraction of the interval [0,1) have you marked? (c) Mark on the interval [0,1) all those points \(x\) which have a digit "1" in at least one position of their base 2 expansion. What fraction of the interval [0,1) have you marked? (d) Mark on the interval [0,1) all those points \(x\) which have a digit "1" in at least one position of their base 3 expansion. What fraction of the interval [0,1) have you marked?

Find numbers \(a\) and \(b\) with the property that the set of solutions of the inequality $$ |x-a|

Solve the following systems of simultaneous equations. (a)(i) \(x+y=1, \quad y+z=2, \quad x+z=3\) (ii) \(u v=2, \quad v w=4, \quad u w=8\) (b) (i) \(x+y=2, \quad y+z=3, \quad x+z=4\) (ii) \(u v=6, \quad v w=10, \quad u w=15\) (iii) \(u v=6, \quad v w=10, \quad u w=30\) (iv) \(u v=4, \quad v w=8, \quad u w=16\)

Let $$ p(x)=x^{2}+\sqrt{2} x+1 $$ Find a polynomial \(q(x)\) such that the product \(p(x) q(x)\) has integer coefficients. $$ \triangle $$
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