Multiply Square Roots

We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says

\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}

We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.

\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}

Remember, we assume all variables are greater than or equal to zero.

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

So we can multiply $\sqrt{3} \cdot \sqrt{5}$ in this way:

\begin{matrix} \sqrt{3} \cdot \sqrt{5} \\ \sqrt{3 \cdot 5} \\ \sqrt{15} \end{matrix}

Sometimes the product gives us a perfect square:

\begin{matrix} \sqrt{2} \cdot \sqrt{8} \\ \sqrt{2 \cdot 8} \\ \sqrt{16} \\ 4 \end{matrix}

Even when the product is not a perfect square, we must look for perfect-square factors 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 $12 x y$ . Keep this in mind as you do these examples.

Simplify: ⓐ $\sqrt{2} \cdot \sqrt{6}$ ⓑ $(4 \sqrt{3}) (2 \sqrt{12})$ .

Solution

ⓐ

This table illustrates the step-by-step process of simplifying the product of square roots, from initial multiplication to the final simplified radical form.
	$\sqrt{2} \cdot \sqrt{6}$
Multiply using the Product Property.	$\sqrt{12}$
Simplify the radical.	$\sqrt{4} \cdot \sqrt{3}$
Simplify.	$2 \sqrt{3}$

ⓑ

This table demonstrates the step-by-step simplification of a radical expression, from multiplication to a final integer result.
	$(4 \sqrt{3}) (2 \sqrt{12})$
Multiply using the Product Property.	$8 \sqrt{36}$
Simplify the radical.	$8 \cdot 6$
Simplify.	$48$

Notice that in (b) we multiplied the coefficients and multiplied the radicals. Also, we did not simplify $\sqrt{12}$ . We waited to get the product and then simplified.

Simplify: $(6 \sqrt{2}) (3 \sqrt{10})$ .

Solution

Demonstration of simplifying a radical expression by multiplication, showing each procedural step and its corresponding mathematical form.
	$(6 \sqrt{2}) (3 \sqrt{10})$
Multiply using the Product Property.	$18 \sqrt{20}$
Simplify the radical.	$18 \sqrt{4} \cdot \sqrt{5}$
Simplify.	$18 \cdot 2 \cdot \sqrt{5}$
	$36 \sqrt{5}$

When we have to multiply square roots, we first find the product and then remove any perfect square factors.

Simplify: ⓐ $(\sqrt{8 x^{3}}) (\sqrt{3 x})$ ⓑ $(\sqrt{20 y^{2}}) (\sqrt{5 y^{3}})$ .

Solution

ⓐ

Illustrates the step-by-step simplification of a product involving square root expressions.
	$(\sqrt{8 x^{3}}) (\sqrt{3 x})$
Multiply using the Product Property.	$\sqrt{24 x^{4}}$
Simplify the radical.	$\sqrt{4 x^{4}} \cdot \sqrt{6}$
Simplify.	$2 x^{2} \sqrt{6}$

ⓑ

Step-by-step simplification of a product of square root expressions.
	$(\sqrt{20 y^{2}}) (\sqrt{5 y^{3}})$
Multiply using the Product Property.	$\sqrt{100 y^{5}}$
Simplify the radical.	$10 y^{2} \sqrt{y}$

Simplify: $(10 \sqrt{6 p^{3}}) (3 \sqrt{18 p})$ .

Solution

Step-by-step simplification of the radical expression (10sqrt(6p^3))(3sqrt(18p)) to its simplified form 180p^2sqrt(3).
	$(10 \sqrt{6 p^{3}}) (3 \sqrt{18 p})$
Multiply.	$30 \sqrt{108 p^{4}}$
Simplify the radical.	$30 \sqrt{36 p^{4}} \cdot \sqrt{3}$
	$30 \cdot 6 p^{2} \cdot \sqrt{3}$
	$180 p^{2} \sqrt{3}$

Simplify: ⓐ ${(\sqrt{2})}^{2}$ ⓑ ${(− \sqrt{11})}^{2}$ .

Solution

ⓐ

Step-by-step simplification of (sqrt(2))^2, illustrating how squaring a square root yields the base number 2.
	${(\sqrt{2})}^{2}$
Rewrite as a product.	$(\sqrt{2}) (\sqrt{2})$
Multiply.	$\sqrt{4}$
Simplify.	$2$

ⓑ

Steps to simplify a squared square root expression, showing rewriting, multiplication, and final integer result.
	${(− \sqrt{11})}^{2}$
Rewrite as a product.	$(− \sqrt{11}) (− \sqrt{11})$
Multiply.	$\sqrt{121}$
Simplify.	$11$

The results of the previous example lead us to this property.

By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify ${(\sqrt{2})}^{2}$ and get 2 right away. When we multiply the two like square roots in part (a) of the next example, it is the same as squaring.

Simplify: ⓐ $(2 \sqrt{3}) (8 \sqrt{3})$ ⓑ ${(3 \sqrt{6})}^{2}$ .

Solution

ⓐ

Step-by-step simplification of the product of radical expressions, (23)(83), to its integer value.
	$(2 \sqrt{3}) (8 \sqrt{3})$
Multiply. Remember, ${(\sqrt{3})}^{2} = 3 .$	$16 \cdot 3$
Simplify.	$48$

ⓑ

Illustrates the step-by-step simplification of the mathematical expression (36) to its final value, 54.
	${(3 \sqrt{6})}^{2}$
Multiply.	$9 \cdot 6$
Simplify.	$54$

Use Polynomial Multiplication to Multiply Square Roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots.

We will first distribute and then simplify the square roots when possible.

Simplify: ⓐ $3 (5 - \sqrt{2})$ ⓑ $\sqrt{2} (4 - \sqrt{10})$ .

Solution

ⓐ

This table illustrates the distributive property by showing an algebraic expression and its expanded form after distribution.
	$3 (5 - \sqrt{2})$
Distribute.	$15 - 3 \sqrt{2}$

ⓑ

Step-by-step simplification of a mathematical expression, illustrating distribution and reduction of square roots.
	$\sqrt{2} (4 - \sqrt{10})$
Distribute.	$4 \sqrt{2} - \sqrt{20}$
	$4 \sqrt{2} - 2 \sqrt{5}$

Simplify: ⓐ $\sqrt{5} (7 + 2 \sqrt{5})$ ⓑ $\sqrt{6} (\sqrt{2} + \sqrt{18})$ .

Solution

ⓐ

This table illustrates the step-by-step simplification of the mathematical expression $\\sqrt{5}(7 + 2\\sqrt{5})$, detailing the multiplication and final simplified form.
	$\sqrt{5} (7 + 2 \sqrt{5})$
Multiply.	$7 \sqrt{5} + 2 \cdot 5$
Simplify.	$7 \sqrt{5} + 10$
	$10 + 7 \sqrt{5}$

ⓑ

Step-by-step simplification of the radical expression sqrt(6)*(sqrt(2) + sqrt(18)).
	$\sqrt{6} (\sqrt{2} + \sqrt{18})$
Multiply.	$\sqrt{12} + \sqrt{108}$
Simplify.	$\sqrt{4} \cdot \sqrt{3} + \sqrt{36} \cdot \sqrt{3}$
	$2 \sqrt{3} + 6 \sqrt{3}$
Combine like radicals.	$8 \sqrt{3}$

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

Solution

Steps to multiply and simplify an expression involving binomials with square roots.
	$(2 + \sqrt{3}) (4 - \sqrt{3})$
Multiply.	$8 - 2 \sqrt{3} + 4 \sqrt{3} - 3$
Combine like terms.	$5 + 2 \sqrt{3}$

Simplify: $(3 - 2 \sqrt{7}) (4 - 2 \sqrt{7})$ .

Solution

Step-by-step multiplication and simplification of an algebraic expression involving 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.	$40 - 14 \sqrt{7}$

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

Solution

This table demonstrates the step-by-step multiplication and simplification of an algebraic expression involving square roots, showing each operation and the resulting expression.
	$(3 \sqrt{2} - \sqrt{5}) (\sqrt{2} + 4 \sqrt{5})$
Multiply.	$3 \cdot 2 + 12 \sqrt{10} - \sqrt{10} - 4 \cdot 5$
Simplify.	$6 + 12 \sqrt{10} - \sqrt{10} - 20$
Combine like terms.	$−14 + 11 \sqrt{10}$

Simplify: $(4 - 2 \sqrt{x}) (1 + 3 \sqrt{x})$ .

Solution

This table demonstrates the step-by-step simplification of the algebraic expression (4-2x)(1+3x) by multiplication and combining like terms.
	$(4 - 2 \sqrt{x}) (1 + 3 \sqrt{x})$
Multiply.	$4 + 12 \sqrt{x} - 2 \sqrt{x} - 6 x$
Combine like terms.	$4 + 10 \sqrt{x} - 6 x$

Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.

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

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 binomial square pattern.

Simplify.

Combine like terms.
ⓑ

Multiply using the binomial square pattern.

Simplify.

Combine like terms.

Simplify: ${(1 + 3 \sqrt{x})}^{2}$ .

Solution


Multiply using the binomial square pattern.
Simplify.

In the next two examples, we will find the product of conjugates.

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

Solution


Multiply using the binomial square pattern.
Simplify.

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

Solution


Multiply using the binomial square pattern.
Simplify.	$A fraction displaying the operation (25 - 4 * 3) divided by 13. Following the order of operations, the numerator simplifies to 25 - 12 = 13, making the fraction 13/13, which equals 1.$

Key Concepts

Product Property of Square Roots If a, b are nonnegative real numbers, then

$\sqrt{a b} = \sqrt{a} \cdot \sqrt{b} and \sqrt{a} \cdot \sqrt{b} = \sqrt{a b}$
Special formulas for multiplying binomials and conjugates:

$\begin{matrix} {(a + b)}^{2} = a^{2} + 2 a b + b^{2} & (a - b) (a + b) = a^{2} - b^{2} \\ {(a - b)}^{2} = a^{2} - 2 a b + b^{2} \end{matrix}$
The FOIL method can be used to multiply binomials containing radicals.

Practice Makes Perfect

Multiply Square Roots

In the following exercises, simplify.

ⓐ $\sqrt{2} \cdot \sqrt{8}$ ⓑ $(3 \sqrt{3}) (2 \sqrt{18})$

Solution

ⓐ $4$ ⓑ $18 \sqrt{6}$

ⓐ $\sqrt{6} \cdot \sqrt{6}$ ⓑ $(3 \sqrt{2}) (2 \sqrt{32})$

ⓐ $\sqrt{7} \cdot \sqrt{14}$ ⓑ $(4 \sqrt{8}) (5 \sqrt{8})$

Solution

ⓐ $7 \sqrt{2}$ ⓑ 160

ⓐ $\sqrt{6} \cdot \sqrt{12}$ ⓑ $(2 \sqrt{5}) (2 \sqrt{10})$

$(5 \sqrt{2}) (3 \sqrt{6})$

Solution

$30 \sqrt{3}$

$(2 \sqrt{3}) (4 \sqrt{6})$

$(−2 \sqrt{3}) (3 \sqrt{18})$

Solution

$−18 \sqrt{6}$

$(−4 \sqrt{5}) (5 \sqrt{10})$

$(5 \sqrt{6}) (− \sqrt{12})$

Solution

$−30 \sqrt{2}$

$(6 \sqrt{2}) (− \sqrt{10})$

$(−2 \sqrt{7}) (−2 \sqrt{14})$

Solution

$28 \sqrt{2}$

$(−2 \sqrt{11}) (−4 \sqrt{22})$

ⓐ $(\sqrt{15 y}) (\sqrt{5 y^{3}})$ ⓑ $(\sqrt{2 n^{2}}) (\sqrt{18 n^{3}})$

Solution

ⓐ $5 y^{2} \sqrt{3}$ ⓑ $6 n^{2} \sqrt{n}$

ⓐ $(\sqrt{14 x^{3}}) (\sqrt{7 x^{3}})$ ⓑ $(\sqrt{3 q^{2}}) (\sqrt{48 q^{3}})$

ⓐ $(\sqrt{16 y^{2}}) (\sqrt{8 y^{4}})$ ⓑ $(\sqrt{11 s^{6}}) (\sqrt{11 s})$

Solution

ⓐ $8 y^{3} \sqrt{2}$ ⓑ $11 s^{3} \sqrt{s}$

ⓐ $(\sqrt{8 x^{3}}) (\sqrt{3 x})$ ⓑ $(\sqrt{7 r}) (\sqrt{7 r^{8}})$

$(2 \sqrt{5 b^{3}}) (4 \sqrt{15 b})$

Solution

$40 b^{2} \sqrt{3}$

$(3 \sqrt{8 c^{5}}) (2 \sqrt{6 c^{3}})$

$(5 \sqrt{2 d^{7}}) (3 \sqrt{50 d^{3}})$

Solution

$150 d^{5}$

$(4 \sqrt{6 t^{2}}) (3 \sqrt{3 t^{2}})$

$(3 \sqrt{4 y^{4}}) (3 \sqrt{9 y^{5}})$

Solution

$54 y^{4} \sqrt{y}$

$(−2 \sqrt{7 z^{3}}) (3 \sqrt{14 z^{8}})$

$(4 \sqrt{2 k^{5}}) (−3 \sqrt{32 k^{6}})$

Solution

$−96 k^{5} \sqrt{k}$

ⓐ ${(\sqrt{7})}^{2}$ ⓑ ${(− \sqrt{15})}^{2}$

ⓐ ${(\sqrt{11})}^{2}$ ⓑ ${(− \sqrt{21})}^{2}$

Solution

ⓐ 11 ⓑ 21

ⓐ ${(\sqrt{19})}^{2}$ ⓑ ${(− \sqrt{5})}^{2}$

ⓐ ${(\sqrt{23})}^{2}$
ⓑ ${(− \sqrt{3})}^{2}$

Solution

ⓐ 23 ⓑ 3

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

ⓐ $(2 \sqrt{13}) (−9 \sqrt{13})$ ⓑ ${(6 \sqrt{5})}^{2}$

Solution

ⓐ −234 ⓑ 180

ⓐ $(−3 \sqrt{12}) (−2 \sqrt{6})$ ⓑ ${(−4 \sqrt{10})}^{2}$

ⓐ $(−7 \sqrt{5}) (−3 \sqrt{10})$ ⓑ ${(−2 \sqrt{14})}^{2}$

Solution

ⓐ $105 \sqrt{2}$ ⓑ 56

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

ⓐ $3 (4 - \sqrt{3})$ ⓑ $\sqrt{2} (4 - \sqrt{6})$

ⓐ $4 (6 - \sqrt{11})$ ⓑ $\sqrt{2} (5 - \sqrt{12})$

Solution

ⓐ $24 - 4 \sqrt{11}$ ⓑ $5 \sqrt{2} - 2 \sqrt{6}$

ⓐ $5 (3 - \sqrt{7})$ ⓑ $\sqrt{3} (4 - \sqrt{15})$

ⓐ $7 (−2 - \sqrt{11})$ ⓑ $\sqrt{7} (6 - \sqrt{14})$

Solution

ⓐ $−14 - 7 \sqrt{11}$ ⓑ $6 \sqrt{7} - 7 \sqrt{2}$

ⓐ $\sqrt{7} (5 + 2 \sqrt{7})$ ⓑ $\sqrt{5} (\sqrt{10} + \sqrt{18})$

ⓐ $\sqrt{11} (8 + 4 \sqrt{11})$ ⓑ $\sqrt{3} (\sqrt{12} + \sqrt{27})$

Solution

ⓐ $44 + 8 \sqrt{11}$ ⓑ $15$

ⓐ $\sqrt{11} (−3 + 4 \sqrt{11})$ ⓑ $\sqrt{3} (\sqrt{15} - \sqrt{18})$

ⓐ $\sqrt{2} (−5 + 9 \sqrt{2})$ ⓑ $\sqrt{7} (\sqrt{3} - \sqrt{21})$

Solution

ⓐ $18 - 5 \sqrt{2}$ ⓑ $\sqrt{21} - 7 \sqrt{3}$

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

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

Solution

$60 + 2 \sqrt{3}$

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

$(9 - \sqrt{2}) (6 + \sqrt{2})$

Solution

$52 + 3 \sqrt{2}$

$(3 - \sqrt{7}) (5 - \sqrt{7})$

$(5 - \sqrt{7}) (4 - \sqrt{7})$

Solution

$27 - 9 \sqrt{7}$

$(1 + 3 \sqrt{10}) (5 - 2 \sqrt{10})$

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

Solution

$−62 + 55 \sqrt{5}$

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

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

Solution

$41 + 7 \sqrt{55}$

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

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

Solution

$−81 + 44 \sqrt{78}$

$(5 - \sqrt{u}) (3 + \sqrt{u})$

$(9 - \sqrt{w}) (2 + \sqrt{w})$

Solution

$18 + 7 \sqrt{w} - w$

$(7 + 2 \sqrt{m}) (4 + 9 \sqrt{m})$

$(6 + 5 \sqrt{n}) (11 + 3 \sqrt{n})$

Solution

$66 + 73 \sqrt{n} + 15 n$

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

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

Solution

ⓐ $27 + 8 \sqrt{11}$ ⓑ $29 - 12 \sqrt{5}$

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

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

Solution

ⓐ $35 - 10 \sqrt{10}$ ⓑ $82 + 48 \sqrt{2}$

$(3 - \sqrt{5}) (3 + \sqrt{5})$

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

Solution

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

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

Solution

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

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

Solution

$−127$

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

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

Solution

Mixed Practice

In the following exercises, simplify.

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

$(4 \sqrt{6}) (− \sqrt{18})$

Solution

$−24 \sqrt{3}$

$(−5 + \sqrt{7}) (6 + \sqrt{21})$

$(−5 \sqrt{7}) (6 \sqrt{21})$

Solution

$−210 \sqrt{3}$

$(−4 \sqrt{2}) (2 \sqrt{18})$

$(\sqrt{35 y^{3}}) (\sqrt{7 y^{3}})$

Solution

$7 y^{3} \sqrt{5}$

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

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

Solution

$29$

$(−4 \sqrt{17}) (−3 \sqrt{17})$

$(−4 + \sqrt{17}) (−3 + \sqrt{17})$

Solution

$29 - 7 \sqrt{17}$

Everyday Math

A landscaper wants to put a square reflecting pool next to a triangular deck, as shown below. The triangular deck is a right triangle, with legs of length 9 feet and 11 feet, and the pool will be adjacent to the hypotenuse.

ⓐ Use the Pythagorean Theorem to find the length of a side of the pool. Round your answer to the nearest tenth of a foot.
ⓑ Find the exact area of the pool.

This figure is an illustration of a square pool with a deck in the shape of a right triangle. the pool's sides are x inches long while the deck's hypotenuse is x inches long and its legs are nine and eleven inches long.

An artist wants to make a small monument in the shape of a square base topped by a right triangle, as shown below. The square base will be adjacent to one leg of the triangle. The other leg of the triangle will measure 2 feet and the hypotenuse will be 5 feet.

ⓐ Use the Pythagorean Theorem to find the length of a side of the square base. Round your answer to the nearest tenth of a foot.
ⓑ Find the exact area of the face of the square base.

Solution

ⓐ $4.6 feet$ ⓑ $21 sq. feet$

A square garden will be made with a stone border on one edge. If only $3 + \sqrt{10}$ feet of stone are available, simplify ${(3 + \sqrt{10})}^{2}$ to determine the area of the largest such garden. Round your answer to the nearest tenth of a foot.

A garden will be made so as to contain two square sections, one section with side length $\sqrt{5} + \sqrt{6}$ yards and one section with side length $\sqrt{2} + \sqrt{3}$ yards. Simplify ${(\sqrt{5} + \sqrt{6})}^{2} + {(\sqrt{2} + \sqrt{3})}^{2}$ to determine the total area of the garden. Round your answer to the nearest tenth.

Solution

$31.9 square yards$

Suppose a third section will be added to the garden in the previous exercise. The third section is to have a width of $\sqrt{432}$ yards. Write an expression that gives the total area of the garden.

Writing Exercises

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

Solution

ⓐ when squaring a negative, it becomes a positive ⓑ since the negative is not included in the parenthesis, it is not squared, and remains negative

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 four columns and three rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no minus I don’t get it!” The rows under the “I can…” column read, “multiply square roots.,” and “use polynomial multiplication to multiply square roots.” The other rows under the other columns are empty.

ⓑ 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?