Problem 61
Question
For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{2}{7}} $$
Step-by-Step Solution
Verified Answer
Question: Simplify the given expression: √(2/7).
Answer: The simplified expression is √(2/7).
1Step 1: Identify the given expression
We are given the following expression to simplify:
$$
\sqrt{\frac{2}{7}}
$$
2Step 2: Rationalizing the denominator
Since the denominator 7 is already a prime number and there is no rational factor we can multiply with to get rid of the square root, our expression is already in the simplest form. Thus, the final answer is:
$$
\sqrt{\frac{2}{7}}
$$
Key Concepts
Rationalize the DenominatorPrime NumbersSquare Roots
Rationalize the Denominator
When working with radical expressions, particularly square roots, one common goal is to ensure that the denominator of a fraction doesn't contain any radicals. This process is known as rationalizing the denominator. To do this, multiply both the numerator and the denominator by a number that will yield a perfect square under the radical in the denominator.
For instance, if you have an expression like \(\sqrt{\frac{1}{3}}\), you can rationalize the denominator by multiplying both the top and bottom of the fraction by \(\sqrt{3}\), making the expression \(\sqrt{\frac{1}{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\). However, if the denominator is a prime number that cannot be made into a perfect square by multiplication with a rational number, as with our exercise \(\sqrt{\frac{2}{7}}\), it is already in its simplest form and cannot be further rationalized.
For instance, if you have an expression like \(\sqrt{\frac{1}{3}}\), you can rationalize the denominator by multiplying both the top and bottom of the fraction by \(\sqrt{3}\), making the expression \(\sqrt{\frac{1}{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\). However, if the denominator is a prime number that cannot be made into a perfect square by multiplication with a rational number, as with our exercise \(\sqrt{\frac{2}{7}}\), it is already in its simplest form and cannot be further rationalized.
Prime Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. It's a key concept in mathematics because prime numbers are the building blocks of all natural numbers - they can be used to decompose any number into a product of prime factors.
This is central to simplifying radical expressions, as seen in prime factorization where numbers are broken down into their prime constituents. In our exercise, the number 7 is prime, which means it cannot be factored further into a product of smaller prime numbers. Recognizing prime numbers allows you to quickly determine whether or not a radical is already in its simplest form and whether rationalizing the denominator is possible or necessary.
This is central to simplifying radical expressions, as seen in prime factorization where numbers are broken down into their prime constituents. In our exercise, the number 7 is prime, which means it cannot be factored further into a product of smaller prime numbers. Recognizing prime numbers allows you to quickly determine whether or not a radical is already in its simplest form and whether rationalizing the denominator is possible or necessary.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. Square roots are represented using the radical symbol \(\sqrt{}\). Simplifying a square root involves finding an equivalent expression that has no radical sign or has a simpler radical.
For example, \(\sqrt{16}\) can be simplified to 4 because 4 times 4 equals 16. In the exercise \(\sqrt{\frac{2}{7}}\), neither 2 nor 7 has a square root that is a whole number, so the expression is already as simple as it can be in terms of square roots. Understanding square roots is crucial when dealing with radical expressions since these often involve simplifying or manipulating square roots to achieve the simplest form.
For example, \(\sqrt{16}\) can be simplified to 4 because 4 times 4 equals 16. In the exercise \(\sqrt{\frac{2}{7}}\), neither 2 nor 7 has a square root that is a whole number, so the expression is already as simple as it can be in terms of square roots. Understanding square roots is crucial when dealing with radical expressions since these often involve simplifying or manipulating square roots to achieve the simplest form.
Other exercises in this chapter
Problem 61
Simplify each expression by performing the indicated operation. $$ (3 y-\sqrt{7 y})^{2} $$
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For the following problems, simplify the expressions. $$ \sqrt{4 x^{2}+32 x+64}+\sqrt{10 x^{2}+80 x+160} $$
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Find each of the following products. $$ \sqrt{y^{3}} \sqrt{y^{5}} \sqrt{y^{2}} $$
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For the following problems, simplify each expressions. $$ \frac{2}{1-\sqrt{a}} $$
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