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Chapter 5: Problem 1

Use the distributive property to help simplify each of the following. \(5 \sqrt{18}-2 \sqrt{2}\)

Short Answer

Expert verified
The expression simplifies to \(13 \sqrt{2}\).

Step by step solution

01

Identify the Radicals

Examine each term and identify any radicals that can be simplified. We have \( \sqrt{18} \) and \( \sqrt{2} \).
02

Simplify Radicals

Simplify \( \sqrt{18} \) by finding the prime factorization: \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2} \).
03

Apply the Distributive Property

Substitute the simplified form of \( \sqrt{18} \) into the expression: \( 5 \times 3 \sqrt{2} - 2 \sqrt{2} \). This becomes \( 15 \sqrt{2} - 2 \sqrt{2} \).
04

Combine Like Terms

Combine the terms with \( \sqrt{2} \) as the common factor: \( (15-2) \sqrt{2} = 13 \sqrt{2} \).

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Key Concepts

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

Radical Simplification
Radical simplification involves breaking down a radical expression into its simplest form. When dealing with radicals, it helps to simplify them to work with easier numbers. Consider the expression \( \sqrt{18} \). To simplify it, you start by performing prime factorization:
  • Break down the number 18 into its prime factors. We find that 18 can be factored as 9 and 2, where 9 is a perfect square \((3^2)\).
  • Write \( \sqrt{18} = \sqrt{9 \times 2} \), which can be separated into \( \sqrt{9} \times \sqrt{2} \).
  • Further simplify \( \sqrt{9} \) to 3, because \( 3^2 = 9 \).
In this way, \( \sqrt{18} \) simplifies to \( 3 \sqrt{2} \). Recognizing and simplifying radicals makes it much easier to combine and manipulate them in expressions.
Prime Factorization
Prime factorization is the process of breaking a number down into its basic building blocks, or prime numbers. It's an essential tool for radical simplification. Let's consider how it was applied to \( \sqrt{18} \):
  • A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
  • Prime factorization of 18 is done by dividing it into the smallest prime numbers: 18 can be expressed as \( 2 \times 3^2 \).
  • This means \( \sqrt{18} \) can be rewritten to \( \sqrt{9 \times 2} \) simplifying it to \( 3 \sqrt{2} \).
Understanding prime factorization is key to reducing complex radicals into simpler forms, facilitating easier calculations and manipulations.
Combining Like Terms
Combining like terms is a fundamental algebraic process that simplifies expressions, especially after distributing. In the given problem, we've gotten to a point where we have:
  • Terms \( 15 \sqrt{2} - 2 \sqrt{2} \) present, which both include the like term \( \sqrt{2} \).
  • Combine them by performing basic arithmetic operations on their coefficients: \( 15 - 2 = 13 \).
  • This results in the combined expression: \( 13 \sqrt{2} \).
Creating unified terms from similar ones not only simplifies expressions but also makes further calculations and understanding much easier. Recognizing and combining these like terms is crucial for completing algebraic problems efficiently.

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Most popular questions from this chapter

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

Use scientific notation and the properties of exponents to help you perform the following operations. \(\sqrt[3]{8000}\)

Your friend keeps getting an error message when evaluating \(-4^{\frac{5}{2}}\) on his calculator. What error is he probably making?

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(y^{\frac{3}{4}}\right)\left(y^{-\frac{1}{2}}\right)\)

In the year 2000 the public debt of the United States was approximately \(\$ 5,700,000,000,000\). For July 2000 , the census reported that \(275,000,000\) people lived in the United States. Convert these figures to scientific notation, and compute the average debt per person. Express the result in scientific notation.
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