5.07 Operations with numerical radicals

We have seen that we can simplify radicals using the property \sqrt{a \cdot b}=\sqrt{a} \cdot \sqrt{b}\,.

If we wanted to multiply two radicals together, we could combine them back in the same way. That is \sqrt{a} \cdot \sqrt{b}=\sqrt{a \cdot b}\,Unlike adding and subtracting radicals, these do not have to be like terms before multiplying.

If we want to find the product of radicals that have a coefficient, we can use the properties of multiplication to rearrange the expression.

Say we have 2\sqrt{5} \cdot 7\sqrt{3}, which we can rewrite as 2\cdot \sqrt{5} \cdot 7 \cdot \sqrt{3}. Using the commutative property, we can also write this expression as 2\cdot 7 \cdot \sqrt{5} \cdot \sqrt{3}. We can simplify this by multiplying the numbers together in one group and the radicals together in another. This becomes 2 \cdot 7 \cdot \sqrt{5} \cdot \sqrt{3} =14 \sqrt{15}

Although simplifying radicals before starting the question is not necessary with multiplication and division like it is with addition and subtraction, it can still help us get the job done quicker sometimes.

Examples

We looked at simplifying radicals which related to multiplication and division, but what about addition and subtraction? Let's explore addition and subtraction of radicals.

Let's look at \sqrt{16}+\sqrt{9}.

16 and 9 are perfect squares, so we can simplify \sqrt{16}+\sqrt{9} to 4+3=7.

Let's look at whether \sqrt{16}+\sqrt{9} is the same as \sqrt{16+9}

\sqrt{16+9} = \sqrt{25} = 5

So we can see that, in general, \sqrt{a}+\sqrt{b} \neq \sqrt{a+b}, and similarly \sqrt{a}-\sqrt{b} \neq \sqrt{a-b}.

So how can we add and subtract radicals? Well unfortunately, if the radicands, a and b, are different values, we can not simplify the expression \sqrt{a}+\sqrt{b}. But if they were the same number, then \\\sqrt{a}+\sqrt{a} = 2\cdot \sqrt{a} = 2\sqrt{a} just like collecting like terms in algebra. We can summarize this as:

Sometimes we are asked to add and subtract radicals that have different radicands (arguments). In this case, we can try to simplify one of the radicals so that we have the same radicands.

When adding and subtracting radicals, they must have the same radicand before we can simplify. Be sure to simplify all radicals first.

Examples

Example 3

Simplify: \sqrt[3]{24}-\sqrt[3]{3}

Worked Solution

Create a strategy

Simplify each radical before subtracting.

Apply the idea

We can find that 24 = 8 \cdot 3 where 8 is a perfect cube.

\displaystyle \sqrt[3]{24}-\sqrt[3]{3}	\displaystyle =	\displaystyle \sqrt[3]{8} \cdot \sqrt[3]{3} - \sqrt[3]{3}	Write the radicals as a products of their factors
	\displaystyle =	\displaystyle 2 \sqrt[3]{3} - \sqrt[3]{3}	Evaluate \sqrt[3]{8}
	\displaystyle =	\displaystyle \sqrt[3]{3}	Evaluate

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Example 4

Fully simplify the expression 5\sqrt{2}+26\sqrt{3}+22\sqrt{3}-8\sqrt{2}.

Worked Solution

Create a strategy

Add like radicals.

Apply the idea

\displaystyle 5\sqrt{2}+26\sqrt{3}+22\sqrt{3}-8\sqrt{2}	\displaystyle =	\displaystyle (5\sqrt{2}-8\sqrt{2})+(26\sqrt{3}+22\sqrt{3})	Group like radicals
	\displaystyle =	\displaystyle 48\sqrt{3}-3\sqrt{2}	Evaluate

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Example 5

Consider the rectangle shown.

a

Find the exact perimeter of the rectangle.

Give your answer in the form a\sqrt{b}, where a and b are integers.

Worked Solution

Create a strategy

Add the four side lengths of the rectangle.

Apply the idea

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 2 (\sqrt{8})+2(5\sqrt{50})	Add the lengths and widths
	\displaystyle =	\displaystyle 2\sqrt{8} + 10\sqrt{50}	Perform multiplication

We can split up both \sqrt{8} and \sqrt{50} into two factors, with one being a perfect square.

\displaystyle 2\sqrt{8} + 10\sqrt{50}	\displaystyle =	\displaystyle 2\sqrt{4} \cdot \sqrt{2} + 10\sqrt{25} \cdot \sqrt{2}	Write the radical as a product of its factors
	\displaystyle =	\displaystyle 2\cdot 2\cdot \sqrt{2}+10\cdot 5\cdot \sqrt{2}	Evaluate \sqrt{4} and \sqrt{25}
	\displaystyle =	\displaystyle 4\sqrt{2}+50\sqrt{2}	Evalute the product of the coefficients
	\displaystyle =	\displaystyle 54\sqrt{2}\text{ cm}	Evaluate

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b

Find the exact area of the rectangle.

Worked Solution

Create a strategy

Use the formula: \text{Area of rectangle} = \text{Length} \cdot \text{Width}

Apply the idea

\displaystyle \text{Area of rectangle}	\displaystyle =	\displaystyle 5\sqrt{50}\cdot \sqrt{8}	Substitute \text{Length}=5\sqrt{50} and \text{Width}=\sqrt{8}
	\displaystyle =	\displaystyle 5\sqrt{4}\cdot \sqrt{2}\cdot \sqrt{25}\cdot \sqrt{2}	Write the radical as a product of its factors
	\displaystyle =	\displaystyle 5\cdot 2\cdot \sqrt{2}\cdot 5\cdot \sqrt{2}	Evaluate \sqrt{4} and \sqrt{25}
	\displaystyle =	\displaystyle 50\sqrt{2}\cdot \sqrt{2}	Evalute the product of the coefficients
	\displaystyle =	\displaystyle 50 \cdot 2	Evaluate the product of the radicals
	\displaystyle =	\displaystyle 100\text{ cm}^2	Evaluate

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