An Introduction to Radicals


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Radicals

Recall

 

Mixed and Entire Radicals

There are two different ways that we can represent a radical.

 

Simplifying Radicals

So far, the questions considered have evaluated nicely: \(\sqrt{25} = 5.\)

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Important Facts

Recall

If \(x\) and \(y\) are positive real numbers, and \(n\) is any real number, then

\((xy)^n=x^ny^n \)

 

Example 3

Simplify \(\sqrt{18}\).

 

Example 4 

Simplify \(\sqrt{192}\).

Example 5

Simplify \(\sqrt[3]{54}\).

 

 

Check Your Understanding 1

Simplify \(\sqrt{(((((((r)*(a))*(d))*(i))*(c))*(a))*(n))*(d)}\).

Enter \(\sqrt{a}\) as 'sqrt\((a)\)'.

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\(\begin{align*} \sqrt{(((((((r)*(a))*(d))*(i))*(c))*(a))*(n))*(d)} &= \sqrt{(((((((a)*(S))*(q))*(u))*(a))*(r))*(e))*(d) \times b} \\ &= \sqrt{(((((((a)*(S))*(q))*(u))*(a))*(r))*(e))*(d)} \times \sqrt{b} \\ &= a\sqrt{b} \end{align*}\)

 

Example 6

Express \(5\sqrt{3}\) as an entire radical.

 

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Check Your Understanding 2


Express \(a\sqrt{b}\) as an entire radical.

Enter \(\sqrt{a}\) as "sqrt\((a)\)".

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How Did I Do?Try Another

Example 7

Simplify \(\sqrt{\dfrac{4}{9}}\).

Solution

Approach 1: Power of a Quotient Rule

\(\sqrt{\dfrac{4}{9}}\)

\(=\left(\dfrac49\right)^\frac12\)

  • Rewritten in exponent form.

 

\(=\dfrac{4^\frac12}{9^\frac12}\)

 

 

\(=\dfrac{\sqrt{4}}{\sqrt{9}}\)

  • Rewritten in radical form.

 

\(=\dfrac23\)

 

Approach 2: Meaning of the Square Root

The square root of \(\dfrac{4}{9}\) is asking us to find the positive number whose product with itself gives \(\dfrac{4}{9}\).

\(\begin{align*} \sqrt{\dfrac{4}{9}}& \; = \sqrt{\dfrac{2}{3}\times\dfrac{2}{3}} \\ & \; = \sqrt{\left (\dfrac{2}{3}\right)^2} \\ & \; =\dfrac23 \end{align*}\)

 

We recognize that \(\dfrac{2}{3}\) has the same sign as \(\dfrac{4}{9}\) (they are both positive), which is what we are looking for in our principle \(n\)th root.

Example 8

Simplify \(\sqrt{\dfrac{147}{108}}\).

Solution

Approach 1

Applying our exponent rules, we can write

\(\sqrt{\dfrac{147}{108}}=\dfrac{\sqrt{147}}{\sqrt{108}}\)

So at this point, we're looking to simplify numerator and denominator individually. That is, we first search for a perfect square factor of \(147\) and then \(108\), to obtain

\( \dfrac{\sqrt{147}}{\sqrt{108}}=\dfrac{\sqrt{49\times3}}{\sqrt{36\times3}}\)

This simplifies to

\(\dfrac{7\sqrt{3}}{6\sqrt{3}}\)

After cancelling the \(\sqrt{3}\) from both the numerator and denominator, we're left with

\(\dfrac{7}{6}\)

Approach 2

As is typically the case when working with fractions, reducing them to lowest terms is generally a great first step.

We might have recognized that the numerator \(147\) has a digit sum of \(12\) which is a multiple of \(3\), so \(147\) is divisible by \(3\). Similarly, \(108\) has a digit sum of \(9\) which is a multiple of \(3\), so \(108\) is also divisible by \(3\).

This gives

\(\sqrt{\dfrac{147}{108}}\)

\(= \sqrt{\dfrac{3\times 49}{3\times 36}}\)

 

 

\(=\sqrt{\dfrac{49}{36}}\)

  • Simplifying.

 

\(=\dfrac{\sqrt{49}}{\sqrt{36}}\)

  • Applying exponent rules.

 

\(=\dfrac76\)

 

Did You Know?

In the above example, the rule for divisibility by \(3\) is used. There are rules that can be used to check for divisibility by other numbers.

Perform an internet search to find Divisibility Rules for other numbers!


Check Your Understanding 3


Simplify \(\sqrt{\dfrac{((n)*(u))*(m)}{((d)*(e))*(n)}}\).

Enter \(\dfrac{a}{b}\) as "\(a/b\)".

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How Did I Do?Try Another

Example 9

Simplify \(\sqrt{30}\).

Solution

Consider factors of \(30\):

\(2\), \(3\), \(5\), \(6\), \(10\), \(15\), \(30\)

Notice the list of factors does not include any of the perfect squares up to \(30\): \(4\), \(9\), \(16\), \(25\).

Therefore, it is not possible to express \(30\) as the product of a perfect square and some other number, and so \(\sqrt{30}\) is in simplified form.


Check Your Understanding 4


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How Did I Do?