In this explainer, we will learn how to multiply and divide square roots and use that to simplify expressions.
We can begin by recalling that is the nonnegative number such that . In particular, this tells us that if , then
This gives us the following result.
Property: Product of the Square Root of a Nonnegative Real Number
If is a nonnegative real number, then .
We can use this property, alongside the commutative and associative properties of multiplication, to simplify expressions involving calculating the product of the square root of a real number with itself. For example,
Let’s now see another example where we need to use this property to simplify an expression.
Example 1: Multiplying Two Radical Numbers with the Same Base
Calculate .
Answer
We can rewrite this expression by using the commutative and associative properties of the multiplication of real numbers as follows:
We can then recall that, for any nonnegative real number , . Hence,
Let’s see another example of using this property to simplify an expression.
Example 2: Multiplying a Radical Number and a Radical Expression with the Same Base
Simplify .
Answer
We start by distributing the product over the parentheses to get
We can simplify this expression by recalling that 1 is the multiplicative identity, so multiplying by 1 leaves the value unchanged, and we can use the associativity of multiplication to rewrite the second term as
We can then simplify this expression by recalling that, for any nonnegative real number , . Hence,
In our next example, we will simplify the product of two radical expressions.
Example 3: Multiplying two Radical Expressions and Simplifying
Express in its simplest form.
Answer
We start by distributing over the parentheses to get
Taking out the factors of and canceling like terms gives
We can reorder the second term and simplify to get
We then rearrange the expressions and evaluate to get
We can then simplify this expression by recalling that, for any nonnegative real number , . Hence,
We have demonstrated how to manipulate and evaluate expressions involving products of square roots. We can then ask, “What happens when we multiply the square roots of any two real numbers?” Of course, we cannot take the square root of negative numbers, so we will begin by assuming that our numbers are nonnegative real numbers, and . We can then note that we know how to simplify and , which we can use to see the following:
We can then take the square root of both sides of the equation. Then, by using the fact that and are nonnegative, we have that
We have therefore shown the following property.
Property: Product of Square Roots of Nonnegative Real Numbers
If and are nonnegative real numbers, then .
We can use this result to simplify expressions involving the product of the square roots of nonnegative real numbers. For example,
We can follow a similar process to evaluate the quotient of two square roots. This time we assume that and are nonnegative and that is nonzero. We can then apply the same reasoning as before to see the following:
Taking square roots of both sides of the equation gives us the following result.
Property: Square Root of the Quotient of Nonnegative Real Numbers
If and are nonnegative real numbers and , then .
We can use this process to simplify expressions involving the quotient of the square roots of nonnegative real numbers. For example,
It is worth noting that both of these results also work in the other direction. For example,
Let’s now see an example of how to use these properties to write an expression involving radicals in its simplest form.
Example 4: Squaring a Radical Expression and Simplifying
Express in its simplest form.
Answer
We first distribute the exponent over the parentheses to get
We can reorder the third term and simplify as follows:
We then recall that if is a nonnegative real number, then . This means that we can simplify this expression as follows:
We can simplify further by recalling that if and are nonnegative real numbers, then . We can apply this in two different ways to this expression.
First,
Second, we can factor the radicand 84 into primes to get . We can use this to simplify the radical:
Hence,
In our next example, we will simplify an expression involving the quotient of two radicals with different bases.
Example 5: Dividing Two Radical Numbers with Different Bases
Express in its simplest form.
Answer
We first recall that if and are nonnegative real numbers and , then . Applying this to the given expression yields
We note that 10 and 2 share a common factor of 2. Canceling this shared factor yields
In our next example, we will simplify an expression using all of the properties we have found in this explainer.
Example 6: Simplifying an Expression Involving the Multiplication and Division of Radicals
Express in its simplest form.
Answer
We start by distributing over the parentheses to get
We then recall that if is a nonnegative real number, then , and if and are nonnegative real numbers, then . Applying this to the expression gives
We then recall that if and are nonnegative real numbers and , then . Applying this we get
We then note that and , so
Let’s finish by recapping some of the important points from this explainer.
Key Points
- If is a nonnegative real number, then .
- If and are nonnegative real numbers, then .
- If and are nonnegative real numbers and , then .
