/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 132 Show that \(\sqrt{2}+\sqrt{72}=\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 132

Show that \(\sqrt{2}+\sqrt{72}=\sqrt{98}\).

Short Answer

Expert verified
To prove that \(\sqrt{2}+\sqrt{72}=\sqrt{98}\), first simplify \(\sqrt{72}\) as \(6\sqrt{2}\). Then, combine the terms to get \(7\sqrt{2}\). Squaring both sides, \((7\sqrt{2})^2\) and \((\sqrt{98})^2\), we find that they are equal (98 = 98), confirming the equality.

Step by step solution

01

Simplify the square roots

First, simplify each square root. The square root of 72 can be simplified by factoring out the largest perfect square, which is 36. Thus, we get: \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6 \times \sqrt{2}\) Now we have: \(\sqrt{2} + 6 \times \sqrt{2}\)
02

Combining like terms

We can combine the two terms with the common \(\sqrt{2}\) factor: \(\sqrt{2} + 6 \times \sqrt{2} = (1 + 6) \times \sqrt{2} = 7\sqrt{2}\)
03

Squaring both sides

To show that this is equal to \(\sqrt{98}\), square both sides: \((7\sqrt{2})^2 = (\sqrt{98})^2\) Expanding both sides: \(49 \times 2 = 98\) \(98 = 98\) Since both sides are equal after squaring, we can conclude: \(\sqrt{2}+\sqrt{72}=\sqrt{98}\)

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 Roots
Square roots are mathematical expressions used to find a value that, when multiplied by itself, results in the original number. For example, \(\sqrt{9}\) equals 3 because 3 times 3 equals 9. Square roots can often be simplified by factoring out perfect squares.

To simplify square roots effectively:
  • Identify the largest perfect square factor of the number under the square root.
  • Rewrite the square root as the product of the square roots of the factors.
  • Calculate the square root of the perfect square, simplifying the expression.
A good illustration is simplifying \(\sqrt{72} = \sqrt{36\times 2} = \sqrt{36}\times\sqrt{2} = 6\sqrt{2}\), where 36 is the largest perfect square factor of 72. Mastering this process makes complex expressions more manageable.
Combining Like Terms
Combining like terms is a fundamental algebraic process, crucial in simplifying expressions. It involves combining terms with the same variables or square roots. This helps to simplify and streamline calculations.

In the expression \(\sqrt{2} + 6\sqrt{2}\), both terms contain the same square root, \(\sqrt{2}\). By treating \(\sqrt{2}\) as a common factor, you can easily combine these terms:
  • Add or subtract the coefficients of the like terms.
  • Keep the common square root factor, in this case, \(\sqrt{2}\).
Therefore, \(\sqrt{2} + 6\sqrt{2} = (1+6)\sqrt{2} = 7\sqrt{2}\). Recognizing and combining like terms effectively transforms complex expressions into simpler ones.
Factoring Perfect Squares
Factoring perfect squares is a technique used to simplify expressions and equations by identifying numbers that can be expressed as the square of another number. This is particularly helpful in simplifying square root expressions.

A perfect square is a number where the square root is an integer. For example, 36 is a perfect square because \(6 \times 6 = 36\). By factoring perfect squares within square roots, you can break down expressions for easier manipulation:
  • Find the perfect square factor within the expression under the root.
  • Express the number as a product of its perfect square factor and another number.
  • Simplify by separating the factors into separate square roots, reducing the perfect square factor.
This operation helped us to simplify \(\sqrt{72}\) to \(6\sqrt{2}\). Factoring perfect squares streamlines work with square roots.
Squaring Both Sides
Squaring both sides is a powerful technique in mathematics giving us a way to solve equations, check equality, or eliminate square roots. By squaring, you ensure the equality holds true and verify complex expressions.

To apply this method:
  • Square each side of the equation to eliminate square roots or to show equivalence.
  • Multiply the squared terms appropriately.
  • Check if resulting values are equal on both sides of the equation.
In our example, after simplifying and combining like terms, we squared both sides of \(7\sqrt{2} = \sqrt{98}\), resulting in \(49 \times 2 = 98\) to confirm their equality. By performing this algebraic twist, you can often simplify the equation-solving process or validate expressions.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Fermat's Last Theorem states that if \(n\) is an integer greater than 2, then there do not exist positive integers \(x, y,\) and \(z\) such that $$ x^{n}+y^{n}=z^{n} . $$ Fermat's Last Theorem was not proved until 1994, although mathematicians had been trying to find a proof for centuries. Use Fermat's Last Theorem to show that if \(n\) is an integer greater than 2 , then there do not exist positive rational numbers \(x\) and \(y\) such that $$ x^{n}+y^{n}=1 $$ [Hint: Use proof by contradiction: Assume there exist rational numbers \(x=\frac{m}{p}\) and \(y=\frac{q}{r}\) such that \(x^{n}+y^{n}=1 ;\) then show that this assumption leads to a contradiction of Fermat's Last Theorem.]

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

Find all real numbers \(x\) that satisfy the indicated equation. $$ x-7 \sqrt{x}+12=0 $$

Sketch the graph of the given function \(f\) on the interval [-1.3,1.3] $$ f(x)=x^{4}-1.5 $$

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=\frac{x^{4}}{81} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.