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Chapter 9: Problem 46

Simplify the radical expression. $$\sqrt{\frac{7}{81}}$$

Short Answer

Expert verified
The simplified form of \(\sqrt{7/81}\) is \(\sqrt{7}/9\).

Step by step solution

01

Break down the radical into separate roots

We can break down the given expression into \(\sqrt{7}/\sqrt{81}\).
02

Simplify the denominator

\(\sqrt{81}\) is equal to \(9\), so the expression becomes \(\sqrt{7}/9\).
03

Simplify the numerator

\(\sqrt{7}\) cannot be simplified further since it is a prime number, so it remains as it is. Thus the simplification of \( \sqrt{7/81} \) is \(\sqrt{7}/9\).

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Key Concepts

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

Understanding Radicals
Radicals are expressions that include the root symbol, often known as the radical sign \(\sqrt{}\). Radicals are a way of indicating roots, such as square roots and cube roots. They're a shortcut to express either the square root or higher roots of a number.
  • Radicals are used to simplify expressions where the root of a number is necessary.
  • The radicand is the number inside the radical symbol, and we aim to simplify the expression under the radical.
Breaking down and simplifying radicals can help in solving complex mathematical problems by achieving the smallest form of the expression possible. For instance, in the expression \(\sqrt{\frac{7}{81}}\), we can simplify by separating it into its numerator and denominator parts, like \(\sqrt{7}/\sqrt{81}\). This helps simplify the expression as much as possible.
Simplifying Square Roots
Square roots are a special type of radical. They are used to find a number which, when multiplied by itself, gives the original value. When simplifying square roots, we look for perfect squares because they simplify cleanly.
  • For example, \(81\) is a perfect square because \(9 \times 9 = 81\), which means \(\sqrt{81} = 9\).
  • If the number is not a perfect square, like \(7\) in our example, it remains under the square root.
When you simplify square root expressions, always try to break them down into factors, checking for perfect squares or simplifying as much as the context allows, without compromising the accuracy of the expression. So for \(\sqrt{7}\), it cannot be simplified further since \(7\) is prime.
Exploring Prime Numbers
Prime numbers are numbers that have only two distinct positive divisors: 1 and themselves. They are the building blocks of all numbers as every number is either prime or can be factored into prime numbers.
  • Some examples of prime numbers include 2, 3, 5, 7, 11, etc.
  • Unlike composite numbers, prime numbers cannot be broken down into smaller factors except 1 and the number itself.
When simplifying radical expressions, identifying prime numbers can be essential. They often signal that the number under the radical cannot be simplified further. This is what happens in our exercise with \(\sqrt{7}\) since \(7\) is a prime, the expression \(\sqrt{7}\) stays as is, contributing directly to the simplified form \(\frac{\sqrt{7}}{9}\). Understanding primality allows for clarity in determining how far an expression can be broken down.

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Most popular questions from this chapter

Vocabulary State the Pythagorean Theorem.

Solve the equation. $$\sqrt{x-2}+1=7$$

Find the distance between the two points. Round your answer to two decimal places, if necessary. $$(3,-2),(4,6)$$

Reasoning Explain why the equation \(\sqrt{x}=-4\) has no solution.

Like Terms Are \(2 \sqrt{2}\) and \(3 \sqrt{2}\) like terms? Explain.
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