Add, Subtract, and Multiply Radical Expressions

Add and Subtract Radical Expressions

Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.

We add and subtract like radicals in the same way we add and subtract like terms. We know that $3 x + 8 x$ is $11 x .$ Similarly we add $3 \sqrt{x} + 8 \sqrt{x}$ and the result is $11 \sqrt{x} .$

Think about adding like terms with variables as you do the next few examples. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms.

Simplify: ⓐ $2 \sqrt{2} - 7 \sqrt{2}$ ⓑ $5 \sqrt[3]{y} + 4 \sqrt[3]{y}$ ⓒ $7 \sqrt[4]{x} - 2 \sqrt[4]{y} .$

Solution

ⓐ

Example illustrating the subtraction of like radical expressions, with step-by-step explanation.
	$2 \sqrt{2} - 7 \sqrt{2}$
Since the radicals are like, we subtract the coefficients.	$−5 \sqrt{2}$

ⓑ

This table demonstrates adding like radicals by combining coefficients, showing an initial expression, the rationale, and the simplified result.
	$5 \sqrt[3]{y} + 4 \sqrt[3]{y}$
Since the radicals are like, we add the coefficients.	$9 \sqrt[3]{y}$

ⓒ

Table presenting the algebraic expression: 7⁴√x - 2⁴√y.
	$7 \sqrt[4]{x} - 2 \sqrt[4]{y}$

The indices are the same but the radicals are different. These are not like radicals. Since the radicals are not like, we cannot subtract them.

For radicals to be like, they must have the same index and radicand. When the radicands contain more than one variable, as long as all the variables and their exponents are identical, the radicands are the same.

Simplify: ⓐ $2 \sqrt{5 n} - 6 \sqrt{5 n} + 4 \sqrt{5 n}$ ⓑ $\sqrt[4]{3 x y} + 5 \sqrt[4]{3 x y} - 4 \sqrt[4]{3 x y} .$

Solution

ⓐ

Step-by-step simplification of a radical expression, showing the process of combining like radicals to reach a final value of zero.
	$2 \sqrt{5 n} - 6 \sqrt{5 n} + 4 \sqrt{5 n}$
Since the radicals are like, we combine them.	$0 \sqrt{5 n}$
Simplify.	$0$

ⓑ

Example demonstrating the combination of like radicals.
	$\sqrt[4]{3 x y} + 5 \sqrt[4]{3 x y} - 4 \sqrt[4]{3 x y}$
Since the radicals are like, we combine them.	$2 \sqrt[4]{3 x y}$

Remember that we always simplify radicals by removing the largest factor from the radicand that is a power of the index. Once each radical is simplified, we can then decide if they are like radicals.

Simplify: ⓐ $\sqrt{20} + 3 \sqrt{5}$ ⓑ $\sqrt[3]{24} - \sqrt[3]{375}$ ⓒ $\frac{1}{2} \sqrt[4]{48} - \frac{2}{3} \sqrt[4]{243} .$

Solution

ⓐ

Step-by-step simplification of a radical expression involving addition.
	$\sqrt{20} + 3 \sqrt{5}$
Simplify the radicals, when possible.	$\sqrt{4} \cdot \sqrt{5} + 3 \sqrt{5}$
	$2 \sqrt{5} + 3 \sqrt{5}$
Combine the like radicals.	$5 \sqrt{5}$

ⓑ

Step-by-step simplification of a radical expression involving the subtraction of cube roots.
	$\sqrt[3]{24} - \sqrt[3]{375}$
Simplify the radicals.	$\sqrt[3]{8} \cdot \sqrt[3]{3} - \sqrt[3]{125} \cdot \sqrt[3]{3}$
	$2 \sqrt[3]{3} - 5 \sqrt[3]{3}$
Combine the like radicals.	$−3 \sqrt[3]{3}$

ⓒ

Step-by-step solution demonstrating the simplification and combination of radical expressions involving fourth roots.
	$\frac{1}{2} \sqrt[4]{48} - \frac{2}{3} \sqrt[4]{243}$
Simplify the radicals.	$\frac{1}{2} \sqrt[4]{16} \cdot \sqrt[4]{3} - \frac{2}{3} \sqrt[4]{81} \cdot \sqrt[4]{3}$
	$\frac{1}{2} \cdot 2 \cdot \sqrt[4]{3} - \frac{2}{3} \cdot 3 \cdot \sqrt[4]{3}$
	$\sqrt[4]{3} - 2 \sqrt[4]{3}$
Combine the like radicals.	$− \sqrt[4]{3}$

In the next example, we will remove both constant and variable factors from the radicals. Now that we have practiced taking both the even and odd roots of variables, it is common practice at this point for us to assume all variables are greater than or equal to zero so that absolute values are not needed. We will use this assumption throughout the rest of this chapter.

Simplify: ⓐ $9 \sqrt{50 m^{2}} - 6 \sqrt{48 m^{2}}$ ⓑ $\sqrt[3]{54 n^{5}} - \sqrt[3]{16 n^{5}} .$

Solution

ⓐ

Step-by-step guide on simplifying a mathematical expression involving radical terms, detailing the breakdown of square roots and explaining why the final terms cannot be combined.
	$9 \sqrt{50 m^{2}} - 6 \sqrt{48 m^{2}}$
Simplify the radicals.	$9 \sqrt{25 m^{2}} \cdot \sqrt{2} - 6 \sqrt{16 m^{2}} \cdot \sqrt{3}$
	$9 \cdot 5 m \cdot \sqrt{2} - 6 \cdot 4 m \cdot \sqrt{3}$
	$45 m \sqrt{2} - 24 m \sqrt{3}$
The radicals are not like and so cannot be combined.

ⓑ

Illustrates the step-by-step simplification of a mathematical expression involving the subtraction of two cube roots.
	$\sqrt[3]{54 n^{5}} - \sqrt[3]{16 n^{5}}$
Simplify the radicals.	$\sqrt[3]{27 n^{3}} \cdot \sqrt[3]{2 n^{2}} - \sqrt[3]{8 n^{3}} \cdot \sqrt[3]{2 n^{2}}$
	$3 n \sqrt[3]{2 n^{2}} - 2 n \sqrt[3]{2 n^{2}}$
Combine the like radicals.	$n \sqrt[3]{2 n^{2}}$

Multiply Radical Expressions

We have used the Product Property of Roots to simplify square roots by removing the perfect square factors. We can use the Product Property of Roots ‘in reverse’ to multiply square roots. Remember, we assume all variables are greater than or equal to zero.

We will rewrite the Product Property of Roots so we see both ways together.

When we multiply two radicals they must have the same index. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible.

Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply $4 x \cdot 3 y$ we multiply the coefficients together and then the variables. The result is 12xy. Keep this in mind as you do these examples.

Simplify: ⓐ $(6 \sqrt{2}) (3 \sqrt{10})$ ⓑ $(−5 \sqrt[3]{4}) (−4 \sqrt[3]{6}) .$

Solution

ⓐ

Step-by-step guide to multiplying and simplifying radical expressions, from (6√2)(3√10) to 36√5.
	$(6 \sqrt{2}) (3 \sqrt{10})$
Multiply using the Product Property.	$18 \sqrt{20}$
Simplify the radical.	$18 \sqrt{4} \cdot \sqrt{5}$
Simplify.	$18 \cdot 2 \cdot \sqrt{5}$
	$36 \sqrt{5}$

ⓑ

Step-by-step multiplication and simplification of cube root expressions.
	$(−5 \sqrt[3]{4}) (−4 \sqrt[3]{6})$
Multiply using the Product Property.	$20 \sqrt[3]{24}$
Simplify the radical.	$20 \sqrt[3]{8} \cdot \sqrt[3]{3}$
Simplify.	$20 \cdot 2 \cdot \sqrt[3]{3}$
	$40 \sqrt[3]{3}$

We follow the same procedures when there are variables in the radicands.

Simplify: ⓐ $(10 \sqrt{6 p^{3}}) (4 \sqrt{3 p})$ ⓑ $(2 \sqrt[4]{20 y^{2}}) (3 \sqrt[4]{28 y^{3}}) .$

Solution

ⓐ

This table illustrates the step-by-step process of multiplying and simplifying a radical expression, showing the intermediate results.
	$(10 \sqrt{6 p^{3}}) (4 \sqrt{3 p})$
Multiply.	$40 \sqrt{18 p^{4}}$
Simplify the radical.	$40 \sqrt{9 p^{4}} \cdot \sqrt{2}$
Simplify.	$40 \cdot 3 p^{2} \cdot \sqrt{2}$
	$120 p^{2} \sqrt{2}$

ⓑ When the radicands involve large numbers, it is often advantageous to factor them in order to find the perfect powers.

Step-by-step solution for multiplying and simplifying radical expressions involving fourth roots.
	$(2 \sqrt[4]{20 y^{2}}) (3 \sqrt[4]{28 y^{3}})$
Multiply.	$6 \sqrt[4]{4 \cdot 5 \cdot 4 \cdot 7 y^{5}}$
Simplify the radical.	$6 \sqrt[4]{16 y^{4}} \cdot \sqrt[4]{35 y^{}}$
Simplify.	$6 \cdot 2 y \sqrt[4]{35 y^{}}$
Multiply.	$12 y \sqrt[4]{35 y^{}}$

Use Polynomial Multiplication to Multiply Radical Expressions

In the next a few examples, we will use the Distributive Property to multiply expressions with radicals. First we will distribute and then simplify the radicals when possible.

Simplify: ⓐ $\sqrt{6} (\sqrt{2} + \sqrt{18})$ ⓑ $\sqrt[3]{9} (5 - \sqrt[3]{18}) .$

Solution

ⓐ

This table outlines the sequential steps involved in simplifying a radical expression, from initial multiplication to combining radicals.
	$\sqrt{6} (\sqrt{2} + \sqrt{18})$
Multiply.	$\sqrt{12} + \sqrt{108}$
Simplify.	$\sqrt{4} \cdot \sqrt{3} + \sqrt{36} \cdot \sqrt{3}$
Simplify.	$2 \sqrt{3} + 6 \sqrt{3}$
Combine like radicals.	$8 \sqrt{3}$

ⓑ

This table illustrates the step-by-step simplification of an algebraic expression involving cube roots through distribution and simplification.
	$\sqrt[3]{9} (5 - \sqrt[3]{18})$
Distribute.	$5 \sqrt[3]{9} - \sqrt[3]{162}$
Simplify.	$5 \sqrt[3]{9} - \sqrt[3]{27} \cdot \sqrt[3]{6}$
Simplify.	$5 \sqrt[3]{9} - 3 \sqrt[3]{6}$

When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.

Simplify: ⓐ $(3 - 2 \sqrt{7}) (4 - 2 \sqrt{7})$ ⓑ $(\sqrt[3]{x} - 2) (\sqrt[3]{x} + 4) .$

Solution

ⓐ

This table illustrates the step-by-step multiplication and simplification of two binomial expressions containing square roots.
	$(3 - 2 \sqrt{7}) (4 - 2 \sqrt{7})$
Multiply.	$12 - 6 \sqrt{7} - 8 \sqrt{7} + 4 \cdot 7$
Simplify.	$12 - 6 \sqrt{7} - 8 \sqrt{7} + 28$
Combine like terms.	$40 - 14 \sqrt{7}$

ⓑ

This table demonstrates the step-by-step multiplication and simplification of an algebraic expression involving cube roots.
	$(\sqrt[3]{x} - 2) (\sqrt[3]{x} + 4)$
Multiply.	$\sqrt[3]{x^{2}} + 4 \sqrt[3]{x} - 2 \sqrt[3]{x} - 8$
Combine like terms.	$\sqrt[3]{x^{2}} + 2 \sqrt[3]{x} - 8$

Simplify: $(3 \sqrt{2} - \sqrt{5}) (\sqrt{2} + 4 \sqrt{5}) .$

Solution

Step-by-step multiplication and simplification of two radical expressions, showing the progression from initial problem to final solution.
	$(3 \sqrt{2} - \sqrt{5}) (\sqrt{2} + 4 \sqrt{5})$
Multiply.	$3 \cdot 2 + 12 \sqrt{10} - \sqrt{10} - 4 \cdot 5$
Simplify.	$6 + 12 \sqrt{10} - \sqrt{10} - 20$
Combine like terms.	$−14 + 11 \sqrt{10}$

Recognizing some special products made our work easier when we multiplied binomials earlier. This is true when we multiply radicals, too. The special product formulas we used are shown here.

We will use the special product formulas in the next few examples. We will start with the Product of Binomial Squares Pattern.

Simplify: ⓐ ${(2 + \sqrt{3})}^{2}$ ⓑ ${(4 - 2 \sqrt{5})}^{2} .$

Solution

Be sure to include the $2 a b$ term when squaring a binomial.

ⓐ


Multiply, using the Product of Binomial Squares Pattern.
Simplify.
Combine like terms.

ⓑ


Multiply, using the Product of Binomial Squares Pattern.
Simplify.

Combine like terms.

In the next example, we will use the Product of Conjugates Pattern. Notice that the final product has no radical.

Simplify: $(5 - 2 \sqrt{3}) (5 + 2 \sqrt{3}) .$

Solution


Multiply, using the Product of Conjugates Pattern.
Simplify.

Key Concepts

Product Property of Roots
- For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n \geq 2$
  $\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ and $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b}$
Special Products
$\begin{matrix} Binomial Squares & Product of Conjugates \\ {(a + b)}^{2} = a^{2} + 2 a b + b^{2} & (a + b) (a - b) = a^{2} - b^{2} \\ {(a - b)}^{2} = a^{2} - 2 a b + b^{2} \end{matrix}$

Practice Makes Perfect

Add and Subtract Radical Expressions

In the following exercises, simplify.

ⓐ $8 \sqrt{2} - 5 \sqrt{2}$ ⓑ $5 \sqrt[3]{m} + 2 \sqrt[3]{m}$ ⓒ $8 \sqrt[4]{m} - 2 \sqrt[4]{m}$

Solution

ⓐ $3 \sqrt{2}$ ⓑ $7 \sqrt[3]{m}$ ⓒ $6 \sqrt[4]{m}$

ⓐ $7 \sqrt{2} - 3 \sqrt{2}$ ⓑ $7 \sqrt[3]{p} + 2 \sqrt[3]{p}$ ⓒ $5 \sqrt[3]{x} - 3 \sqrt[3]{x}$

ⓐ $3 \sqrt{5} + 6 \sqrt{5}$ ⓑ $9 \sqrt[3]{a} + 3 \sqrt[3]{a}$ ⓒ $5 \sqrt[4]{2 z} + \sqrt[4]{2 z}$

Solution

ⓐ $9 \sqrt{5}$ ⓑ $12 \sqrt[3]{a}$ ⓒ $6 \sqrt[4]{2 z}$

ⓐ $4 \sqrt{5} + 8 \sqrt{5}$ ⓑ $\sqrt[3]{m} - 4 \sqrt[3]{m}$ ⓒ $\sqrt{n} + 3 \sqrt{n}$

ⓐ $3 \sqrt{2 a} - 4 \sqrt{2 a} + 5 \sqrt{2 a}$ ⓑ $5 \sqrt[4]{3 a b} - 3 \sqrt[4]{3 a b} - 2 \sqrt[4]{3 a b}$

Solution

ⓐ $4 \sqrt{2 a}$ ⓑ 0

ⓐ $\sqrt{11 b} - 5 \sqrt{11 b} + 3 \sqrt{11 b}$ ⓑ $8 \sqrt[4]{11 c d} + 5 \sqrt[4]{11 c d} - 9 \sqrt[4]{11 c d}$

ⓐ $8 \sqrt{3 c} + 2 \sqrt{3 c} - 9 \sqrt{3 c}$ ⓑ $2 \sqrt[3]{4 p q} - 5 \sqrt[3]{4 p q} + 4 \sqrt[3]{4 p q}$

Solution

ⓐ $\sqrt{3 c}$ ⓑ $\sqrt[3]{4 p q}$

ⓐ $3 \sqrt{5 d} + 8 \sqrt{5 d} - 11 \sqrt{5 d}$ ⓑ $11 \sqrt[3]{2 r s} - 9 \sqrt[3]{2 r s} + 3 \sqrt[3]{2 r s}$

ⓐ $\sqrt{27} - \sqrt{75}$ ⓑ $\sqrt[3]{40} - \sqrt[3]{320}$ ⓒ $\frac{1}{2} \sqrt[4]{32} + \frac{2}{3} \sqrt[4]{162}$

Solution

ⓐ $−2 \sqrt{3}$ ⓑ $−2 \sqrt[3]{5}$ ⓒ $3 \sqrt[4]{2}$

ⓐ $\sqrt{72} - \sqrt{98}$ ⓑ $\sqrt[3]{24} + \sqrt[3]{81}$ ⓒ $\frac{1}{2} \sqrt[4]{80} - \frac{2}{3} \sqrt[4]{405}$

ⓐ $\sqrt{48} + \sqrt{27}$ ⓑ $\sqrt[3]{54} + \sqrt[3]{128}$ ⓒ $6 \sqrt[4]{5} - \frac{3}{2} \sqrt[4]{80}$

Solution

ⓐ $7 \sqrt{3}$ ⓑ $7 \sqrt[3]{2}$ ⓒ $3 \sqrt[4]{5}$

ⓐ $\sqrt{45} + \sqrt{80}$ ⓑ $\sqrt[3]{81} - \sqrt[3]{192}$ ⓒ $\frac{5}{2} \sqrt[4]{80} + \frac{7}{3} \sqrt[4]{405}$

ⓐ $\sqrt{72 a^{5}} - \sqrt{50 a^{5}}$ ⓑ $9 \sqrt[4]{80 p^{4}} - 6 \sqrt[4]{405 p^{4}}$

Solution

ⓐ $a^{2} \sqrt{2 a}$ ⓑ 0

ⓐ $\sqrt{48 b^{5}} - \sqrt{75 b^{5}}$ ⓑ $8 \sqrt[3]{64 q^{6}} - 3 \sqrt[3]{125 q^{6}}$

ⓐ $\sqrt{80 c^{7}} - \sqrt{20 c^{7}}$ ⓑ $2 \sqrt[4]{162 r^{10}} + 4 \sqrt[4]{32 r^{10}}$

Solution

ⓐ $2 c^{3} \sqrt{5 c}$ ⓑ $14 r^{2} \sqrt[4]{2 r^{2}}$

ⓐ $\sqrt{96 d^{9}} - \sqrt{24 d^{9}}$ ⓑ $5 \sqrt[4]{243 s^{6}} + 2 \sqrt[4]{3 s^{6}}$

$3 \sqrt{128 y^{2}} + 4 y \sqrt{162} - 8 \sqrt{98 y^{2}}$

Solution

$4 y \sqrt{2}$

$3 \sqrt{75 y^{2}} + 8 y \sqrt{48} - \sqrt{300 y^{2}}$

Multiply Radical Expressions

In the following exercises, simplify.

ⓐ $(−2 \sqrt[]{3}) (3 \sqrt[]{18})$ ⓑ $(8 \sqrt[3]{4}) (−4 \sqrt[3]{18})$

Solution

ⓐ $−18 \sqrt{6}$ ⓑ $−64 \sqrt[3]{9}$

ⓐ $(−4 \sqrt[]{5}) (5 \sqrt[]{10})$ ⓑ $(−2 \sqrt[3]{9}) (7 \sqrt[3]{9})$

ⓐ $(5 \sqrt[]{6}) (− \sqrt[]{12})$ ⓑ $(−2 \sqrt[4]{18}) (− \sqrt[4]{9})$

Solution

ⓐ $−30 \sqrt{2}$ ⓑ $6 \sqrt[4]{2}$

ⓐ $(−2 \sqrt{7}) (−2 \sqrt{14})$ ⓑ $(−3 \sqrt[4]{8}) (−5 \sqrt[4]{6})$

ⓐ $(4 \sqrt{12 z^{3}}) (3 \sqrt{9 z})$ ⓑ $(5 \sqrt[3]{3 x^{3}}) (3 \sqrt[3]{18 x^{3}})$

Solution

ⓐ $72 z^{2} \sqrt{3}$ ⓑ $45 x^{2} \sqrt[3]{2}$

ⓐ $(3 \sqrt{2 x^{3}}) (7 \sqrt{18 x^{2}})$ ⓑ $(−6 \sqrt[3]{20 a^{2}}) (−2 \sqrt[3]{16 a^{3}})$

ⓐ $(−2 \sqrt{7 z^{3}}) (3 \sqrt{14 z^{8}})$ ⓑ $(2 \sqrt[4]{8 y^{2}}) (−2 \sqrt[4]{12 y^{3}})$

Solution

ⓐ $−42 z^{5} \sqrt{2 z}$ ⓑ $−8 y \sqrt[4]{6 y}$

ⓐ $(4 \sqrt{2 k^{5}}) (−3 \sqrt{32 k^{6}})$ ⓑ $(− \sqrt[4]{6 b^{3}}) (3 \sqrt[4]{8 b^{3}})$

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ $\sqrt{7} (5 + 2 \sqrt{7})$ ⓑ $\sqrt[3]{6} (4 + \sqrt[3]{18})$

Solution

ⓐ $14 + 5 \sqrt{7}$ ⓑ $4 \sqrt[3]{6} + 3 \sqrt[3]{4}$

ⓐ $\sqrt{11} (8 + 4 \sqrt{11})$ ⓑ $\sqrt[3]{3} (\sqrt[3]{9} + \sqrt[3]{18})$

ⓐ $\sqrt{11} (−3 + 4 \sqrt{11})$ ⓑ $\sqrt[4]{3} (\sqrt[4]{54} + \sqrt[4]{18})$

Solution

ⓐ $44 - 3 \sqrt{11}$ ⓑ $3 \sqrt[4]{2} + \sqrt[4]{54}$

ⓐ $\sqrt{2} (−5 + 9 \sqrt{2})$ ⓑ $\sqrt[4]{2} (\sqrt[4]{12} + \sqrt[4]{24})$

$(7 + \sqrt{3}) (9 - \sqrt{3})$

Solution

$60 + 2 \sqrt{3}$

$(8 - \sqrt{2}) (3 + \sqrt{2})$

ⓐ $(9 - 3 \sqrt{2}) (6 + 4 \sqrt{2})$ ⓑ $(\sqrt[3]{x} - 3) (\sqrt[3]{x} + 1)$

Solution

ⓐ $30 + 18 \sqrt{2}$ ⓑ $\sqrt[3]{x^{2}} - 2 \sqrt[3]{x} - 3$

ⓐ $(3 - 2 \sqrt{7}) (5 - 4 \sqrt{7})$ ⓑ $(\sqrt[3]{x} - 5) (\sqrt[3]{x} - 3)$

ⓐ $(1 + 3 \sqrt{10}) (5 - 2 \sqrt{10})$ ⓑ $(2 \sqrt[3]{x} + 6) (\sqrt[3]{x} + 1)$

Solution

ⓐ $−55 + 13 \sqrt{10}$
ⓑ $2 \sqrt[3]{x^{2}} + 8 \sqrt[3]{x} + 6$

ⓐ $(7 - 2 \sqrt{5}) (4 + 9 \sqrt{5})$ ⓑ $(3 \sqrt[3]{x} + 2) (\sqrt[3]{x} - 2)$

$(\sqrt{3} + \sqrt{10}) (\sqrt{3} + 2 \sqrt{10})$

Solution

$23 + 3 \sqrt{30}$

$(\sqrt{11} + \sqrt{5}) (\sqrt{11} + 6 \sqrt{5})$

$(2 \sqrt{7} - 5 \sqrt{11}) (4 \sqrt{7} + 9 \sqrt{11})$

Solution

$−439 - 2 \sqrt{77}$

$(4 \sqrt{6} + 7 \sqrt{13}) (8 \sqrt{6} - 3 \sqrt{13})$

ⓐ ${(3 + \sqrt{5})}^{2}$ ⓑ ${(2 - 5 \sqrt{3})}^{2}$

Solution

ⓐ $14 + 6 \sqrt{5}$ ⓑ $79 - 20 \sqrt{3}$

ⓐ ${(4 + \sqrt{11})}^{2}$ ⓑ ${(3 - 2 \sqrt{5})}^{2}$

ⓐ ${(9 - \sqrt{6})}^{2}$ ⓑ ${(10 + 3 \sqrt{7})}^{2}$

Solution

ⓐ $87 - 18 \sqrt{6}$
ⓑ $163 + 60 \sqrt{7}$

ⓐ ${(5 - \sqrt{10})}^{2}$ ⓑ ${(8 + 3 \sqrt{2})}^{2}$

$(4 + \sqrt{2}) (4 - \sqrt{2})$

Solution

$(7 + \sqrt{10}) (7 - \sqrt{10})$

$(4 + 9 \sqrt{3}) (4 - 9 \sqrt{3})$

Solution

$−227$

$(1 + 8 \sqrt{2}) (1 - 8 \sqrt{2})$

$(12 - 5 \sqrt{5}) (12 + 5 \sqrt{5})$

Solution

$19$

$(9 - 4 \sqrt{3}) (9 + 4 \sqrt{3})$

$(\sqrt[3]{3 x} + 2) (\sqrt[3]{3 x} - 2)$

Solution

$\sqrt[3]{9 x^{2}} - 4$

$(\sqrt[3]{4 x} + 3) (\sqrt[3]{4 x} - 3)$

Mixed Practice

$\frac{2}{3} \sqrt{27} + \frac{3}{4} \sqrt{48}$

Solution

$5 \sqrt{3}$

$\sqrt{175 k^{4}} - \sqrt{63 k^{4}}$

$\frac{5}{6} \sqrt{162} + \frac{3}{16} \sqrt{128}$

Solution

$9 \sqrt{2}$

$\sqrt[3]{24} + \sqrt[3]{/ 81}$

$\frac{1}{2} \sqrt[4]{80} - \frac{2}{3} \sqrt[4]{405}$

Solution

$− \sqrt[4]{5}$

$8 \sqrt[4]{13} - 4 \sqrt[4]{13} - 3 \sqrt[4]{13}$

$5 \sqrt{12 c^{4}} - 3 \sqrt{27 c^{6}}$

Solution

$10 c^{2} \sqrt{3} - 9 c^{3} \sqrt{3}$

$\sqrt{80 a^{5}} - \sqrt{45 a^{5}}$

$\frac{3}{5} \sqrt{75} - \frac{1}{4} \sqrt{48}$

Solution

$2 \sqrt{3}$

$21 \sqrt[3]{9} - 2 \sqrt[3]{9}$

$8 \sqrt[3]{64 q^{6}} - 3 \sqrt[3]{125 q^{6}}$

Solution

$17 q^{2}$

$11 \sqrt{11} - 10 \sqrt{11}$

$\sqrt{3} \cdot \sqrt{21}$

Solution

$3 \sqrt{7}$

$(4 \sqrt{6}) (− \sqrt{18})$

$(7 \sqrt[3]{4}) (−3 \sqrt[3]{18})$

Solution

$−42 \sqrt[3]{9}$

$(4 \sqrt{12 x^{5}}) (2 \sqrt{6 x^{3}})$

${(\sqrt{29})}^{2}$

Solution

$(−4 \sqrt{17}) (−3 \sqrt{17})$

$(−4 + \sqrt{17}) (−3 + \sqrt{17})$

Solution

$29 - 7 \sqrt{17}$

$(3 \sqrt[4]{8 a^{2}}) (\sqrt[4]{12 a^{3}})$

${(6 - 3 \sqrt{2})}^{2}$

Solution

$54 - 36 \sqrt{2}$

$\sqrt{3} (4 - 3 \sqrt{3})$

$\sqrt[3]{3} (2 \sqrt[3]{9} + \sqrt[3]{18})$

Solution

$6 + 3 \sqrt[3]{2}$

$(\sqrt{6} + \sqrt{3}) (\sqrt{6} + 6 \sqrt{3})$

Writing Exercises

Explain when a radical expression is in simplest form.

Solution

Answers will vary.

Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.

ⓐ Explain why ${(− \sqrt{n})}^{2}$ is always non-negative, for $n \geq 0 .$
ⓑ Explain why $- {(\sqrt{n})}^{2}$ is always non-positive, for $n \geq 0.$

Solution

Answers will vary.

Use the binomial square pattern to simplify ${(3 + \sqrt{2})}^{2} .$ Explain all your steps.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “add and subtract radical expressions.”, “ multiply radical expressions”, and “use polynomial multiplication to multiply radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

like radicals

Like radicals are radical expressions with the same index and the same radicand.