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Chapter 5: Problem 15

Use the distributive property to help simplify each of the following. \(\frac{3 \sqrt{18}}{5}-\frac{5 \sqrt{72}}{6}+\frac{3 \sqrt{98}}{4}\)

Short Answer

Expert verified
The simplified expression is \( \frac{373\sqrt{2}}{60} \).

Step by step solution

01

Express Square Roots in Simplest Form

First, simplify each square root expression: - For \( \sqrt{18} \), notice that \( 18 = 9 \times 2 \). Thus, \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \).- For \( \sqrt{72} \), notice that \( 72 = 36 \times 2 \). Thus, \( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \).- For \( \sqrt{98} \), notice that \( 98 = 49 \times 2 \). Thus, \( \sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2} \).
02

Substitute Simplified Expressions

Substitute the simplified forms into the initial expression:\[ \frac{3 \sqrt{18}}{5} - \frac{5 \sqrt{72}}{6} + \frac{3 \sqrt{98}}{4} \] becomes:\[ \frac{3 \times 3\sqrt{2}}{5} - \frac{5 \times 6\sqrt{2}}{6} + \frac{3 \times 7\sqrt{2}}{4} \].
03

Simplify Each Term

Compute the coefficients for each term:- For the first term \( \frac{3 \times 3\sqrt{2}}{5} \), this simplifies to \( \frac{9\sqrt{2}}{5} \).- For the second term \( \frac{5 \times 6\sqrt{2}}{6} \), cancel out the factor of 6, simplifying to \( 5\sqrt{2} \).- For the third term \( \frac{3 \times 7\sqrt{2}}{4} \), this simplifies to \( \frac{21\sqrt{2}}{4} \).
04

Make Denominators Common

To simplify the expression further, find a common denominator, which is 60 for denominators 5, 6, and 4.- Multiply \( \frac{9\sqrt{2}}{5} \) by \( \frac{12}{12} \) resulting in \( \frac{108\sqrt{2}}{60} \).- Multiply \( 5\sqrt{2} \) by \( \frac{10}{10} \) resulting in \( \frac{50\sqrt{2}}{60} \).- Multiply \( \frac{21\sqrt{2}}{4} \) by \( \frac{15}{15} \) resulting in \( \frac{315\sqrt{2}}{60} \).
05

Combine the Expressions

Combine all the terms into a single expression by adding the numerators:\[ \frac{108\sqrt{2}}{60} - \frac{50\sqrt{2}}{60} + \frac{315\sqrt{2}}{60} = \frac{(108 - 50 + 315)\sqrt{2}}{60} \].
06

Simplify the Result

Calculate the final expression: Add up the numerators: \[ 108 - 50 + 315 = 373 \].So the expression becomes: \[ \frac{373\sqrt{2}}{60} \].This is already in its simplest form.

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Key Concepts

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

Simplifying Square Roots
Simplifying square roots means finding a simpler or more convenient form of square root expressions. Often, numbers under a square root (\(\sqrt{}\)) can be broken into smaller factors. If a factor is a perfect square, it can be extracted from the square root.
  • For example, consider \(\sqrt{18}\). Instead of calculating the square root directly, notice that 18 is made up of \(9 \times 2\).
  • Since \(\sqrt{9}\) is 3, we can rewrite \(\sqrt{18}\) as \(3\sqrt{2}\).
This method applies to any number by factoring it until one of the factors is a perfect square:
  • 72 can be factored to \(36 \times 2\), and since \(\sqrt{36}\) is 6, \(\sqrt{72}\) simplifies to \(6\sqrt{2}\).
  • Similarly, \(98 = 49 \times 2\) simplifies to \(7\sqrt{2}\) because \(\sqrt{49}\) is 7.
Understanding how to break numbers down into their factors makes simplifying square roots easier and improves comprehension of algebraic expressions.
Simplifying Algebraic Expressions
An algebraic expression like \(\frac{3 \times 3\sqrt{2}}{5}\) includes variables, coefficients, and sometimes roots. Simplifying these expressions involves a few important steps.
Start by carrying out operations within the numerator and the denominator separately:
  • The expression \(3 \times 3\sqrt{2}\) simplifies to \(9\sqrt{2}\).
  • Place the result over the existing denominator, in this case, 5.
Combining and simplifying complex fractions becomes straightforward with organized steps such as combining like terms:
  • The original task had the expression broken down into smaller, simpler parts before dealing with it as a whole.
As you deal with coefficients or roots, keep an eye out for any common factors that can be reduced. Consistently practice these techniques to recognize patterns and solutions faster.
Common Denominators in Fractions
Combining fractions involves making the denominators match. This is essential when you add or subtract fractions. If the denominators are different, planning is needed to find a common denominator, which is typically the lowest common multiple (LCM) of the denominators.
  • For example, working with denominators 5, 6, and 4, the LCM is 60.
  • Change each fraction like \(\frac{9\sqrt{2}}{5}\) by multiplying the numerator and the denominator by the necessary amount鈥擻(\frac{12}{12}\) in this case鈥攖o get a new denominator of 60.
After adjusting all fractions to a common denominator, combine by simply handling the numerators, treating the expressions like regular numbers:
  • Sum or subtract the adjusted numerators and place the result over the common denominator.
Mastering this technique simplifies calculations involving multiple fractions and eliminates errors, making working with algebraic expressions more intuitive.

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Most popular questions from this chapter

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(0.00000082\)

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((3.14)(10)^{10}\)

Sometimes it is more convenient to express a number as a product of a power of 10 and a number that is not between 1 and 10 . For example, suppose that we want to calculate \(\sqrt{640,000}\). We can proceed as follows: $$ \begin{aligned} \sqrt{640,000} &=\sqrt{(64)(10)^{4}} \\ &=\left((64)(10)^{4}\right)^{\frac{1}{2}} \\ &=(64)^{\frac{1}{2}}\left(10^{4}\right)^{\frac{1}{2}} \\ &=(8)(10)^{2} \\ &=8(100)=800 \end{aligned} $$ Compute each of the following without a calculator, and then use a calculator to check your answers. (a) \(\sqrt{49,000,000}\) (b) \(\sqrt{0.0025}\) (c) \(\sqrt{14,400}\) (d) \(\sqrt{0.000121}\) (e) \(\sqrt[3]{27,000}\) (f) \(\sqrt[3]{0.000064}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(2 x^{\frac{1}{3}}\right)\left(x^{-\frac{1}{2}}\right)\)

Why do we need scientific notation even when using calculators and computers?
See all solutions

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