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High School Core Standards - Algebra I Assessment Anchors

2.06 Square and cube roots

Lesson

Evaluating square and cube roots

The square root function reverses the squaring function. Similarly, the cube root function undoes the cubing function.

In algebraic notation, when $a\ge0$a0, and $b\ge0$b0, we write $\sqrt{a^2}=a$a2=a and $\sqrt[3]{b^3}=b$3b3=b. It is also true that $\left(\sqrt{a}\right)^2=a$(a)2=a and $\left(\sqrt[3]{b}\right)^3=b$(3b)3=b.

Square root and cube root expressions can sometimes be written in simpler forms using these facts, together with the other familiar radical rules.

We can use these ideas to simplify expressions like $\sqrt{16}$16 and $\sqrt{121}$121, where the numbers in the radical are perfect squares, or expressions like $\sqrt[3]{27}$327 and $\sqrt[3]{125}$3125, where the numbers in the radical are perfect cubes.

 

Worked example

Question 1

Simplify$\frac{\sqrt{64}}{\sqrt{100}}$64100.

Think: We can first evaluate the square roots and then simplify if possible.

Do: It is recommended to memorize the first 12 perfect squares. We should notice that both $64$64 and $100$100 are perfect squares.

$\frac{\sqrt{64}}{\sqrt{100}}$64100 $=$= $\frac{\sqrt{8^2}}{\sqrt{10^2}}$82102
  $=$= $\frac{8}{10}$810
  $=$= $\frac{4}{5}$45

 

Practice questions

Question 2

Simplify $\frac{64}{\sqrt{64}}$6464.

Question 3

Simplify the following algebraic radical: $\sqrt{64x}$64x

 

Square roots and negative numbers

So what happens when we bring in negative numbers?

Firstly, at this stage, we will not have to find the square root of a negative number. Why?

Remember when we were squaring numbers- multiplying two negative numbers always gives a positive answer, so we never get a negative answer. When we take the square root of a negative number we need what is called an imaginary number.

Let's look at questions where we can include a negative symbol without requiring imaginary numbers.

Worked example

Question 4

Evaluate: $-\sqrt{64}$64

Think: This is really $-1\times\sqrt{64}$1×64, so the order of operations tells us to deal with the exponent (or root) first. Also, we know that $8^2=64$82=64.

Do: $-\sqrt{64}=-8$64=8 

 

Cube root of a Negative Number

We can find the cube root of a negative number (because when we cube a negative number, we get a negative answer).

Worked examples

Question 5

Evaluate: $-\sqrt[3]{-216}$3216

Think: $-\sqrt[3]{-216}$3216 is really $-1\times\sqrt[3]{-216}$1×3216, so we need to first evaluate $\sqrt[3]{-216}$3216. We are looking for a number, that when multiplied by itself $3$3 times gives us $-216$216. We can find that $\left(-6\right)^3=-216$(6)3=216.

Do:

$-\sqrt[3]{-216}$3216 $=$= $-1\times\left(-6\right)$1×(6)
  $=$= $6$6

 

Further questions with roots

Building on the concepts we have looked at already, including how to add and subtract integers, how to multiply and divide integers, as well as order of operations, we can do more complex questions using negative square and cube roots.

Worked examples

Question 6

Evaluate: $\sqrt[3]{-64}\times\sqrt{64}$364×64 

Think: The cube root of $-64$64 is $-4$4 and the square root of $64$64 is $8$8.

Do:

$\sqrt[3]{-64}\times\sqrt{64}$364×64 $=$= $-4\times8$4×8
  $=$= $-32$32

 

Practice questions

Question 7

Evaluate $\sqrt{\left(-3\right)^2+4^2}$(3)2+42

Question 8

Evaluate $\sqrt[3]{-125}\times\sqrt[3]{27}$3125×327

 

Simplifying square and cube roots

A radical is can be transformed due to some interesting properties of roots. Let's take a look at square roots for example.

We know that $\sqrt{36}=6$36=6. We can use this fact to understand another property. 

$\sqrt{36}$36 can be written as $\sqrt{4\times9}$4×9, we want to see if this means that $\sqrt{36}=\sqrt{4}\times\sqrt{9}$36=4×9. We know that $\sqrt{4}=2$4=2 and $\sqrt{9}=3$9=3 so $\sqrt{36}$36 would also be rewritten as $2\times3=6$2×3=6, which is the fact we already knew.

We can do this because of a special property:

$\sqrt{a\times b}=\sqrt{a}\times\sqrt{b}$a×b=a×b, where $a$a and $b$b are positive integers

 

Most of the time, $\sqrt{a}$a is a radical that can not be expressed as a rational number. If this the case, we can use the property above to help transform larger radicals into smaller ones.

Did you know?

With the property that $\sqrt{a\times b}=\sqrt{a}\times\sqrt{b}$a×b=a×b, we can factor a number into two factors where one is a perfect square and then simplify the radical.

 

Worked example

Question 9

Simplify $\sqrt{20}$20.

Think: We want to write 20 as a product of two numbers, where one is a perfect square. Then we can simplify the radical.

Do:

$\sqrt{20}$20 $=$= $\sqrt{4\times5}$4×5
  $=$= $\sqrt{4}\times\sqrt{5}$4×5
  $=$= $2\times\sqrt{5}$2×5
  $=$= $2\sqrt{5}$25

Reflect$2\sqrt{5}$25 and this is called a simplified radical, as we can not break it down any more into even smaller radicals.

 

Question 10

Simplify $\sqrt{16a^{10}b^5}$16a10b5.

Think: We can try to rewrite each factor as something squared in order to see what the square root of each factor would be. When we can't rewrite as a perfect square we can factor out the largest possible perfect square.

Do: It is recommended to memorize the first 12 perfect squares. We should notice that $16$16 is a perfect square.

$\sqrt{16a^{10}b^6}$16a10b6 $=$= $\sqrt{4^2\left(a^5\right)^2\left(b^2\right)^2b}$42(a5)2(b2)2b
  $=$= $\sqrt{4^2}\sqrt{\left(a^5\right)^2}\sqrt{\left(b^2\right)^2b}$42(a5)2(b2)2b
  $=$= $4a^5b^2\sqrt{b}$4a5b2b

 

Exploration

One method to check whether an expression can be simplified is by looking at its prime factors. For example, given the expression $\sqrt{18}$18, we first look at the prime factors of $18$18 which gives us $3\times3\times2=3^2\times2$3×3×2=32×2.

Now we can write $\sqrt{18}$18 as $\sqrt{3^2\times2}=\sqrt{3^2}\times\sqrt{2}$32×2=32×2. In this step we have used the important fact that $\sqrt{a\times b}=\sqrt{a}\times\sqrt{b}$a×b=a×b.

The first term in the product $\sqrt{3^2}\times\sqrt{2}$32×2 is of the form $\sqrt{a^2}=a$a2=a, so the fully simplified expression becomes $3\sqrt{2}$32.

From this example, we can see that if any factor appears two times within a square root or three times within a cube root, then the expression can be further simplified. This is equivalent to looking for a factor that is a perfect square for square root expressions, or a perfect cube for cube root expressions.

 

Simplifying With Fractions

Just as there is a square root property involving multiplication, there is also a similar property involving division:

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ab=ab

Again we can use this to help us transform and simplify radicals. For example $\frac{\sqrt{8}}{\sqrt{2}}$82 can be rewritten as $\sqrt{\frac{8}{2}}=\sqrt{4}$82=4 which simplifies to $2$2.

 

Worked example

Question 11

Simplify $\sqrt{52}\div\sqrt{13}$52÷​13.

Think: remember that a division operation can be rewritten as a fraction

Do: 

$\sqrt{52}\div\sqrt{13}$52÷​13 $=$= $\frac{\sqrt{52}}{\sqrt{13}}$5213
  $=$= $\sqrt{\frac{52}{13}}$5213
  $=$= $\sqrt{4}$4
  $=$= $2$2

 

Squaring Radicals

If we were to look at $\sqrt{36}$36 again, and we decided to square it, what answer would we get? $\left(\sqrt{36}\right)^2$(36)2 = $6^2$62 = $36$36. We've come full circle and back to $36$36! This brings us to the third property of square roots:

$\left(\sqrt{a}\right)^2$(a)2 = $a$a

 

This makes sense as the square root and square operations are 'opposite' or inverse operations to each other like addition and subtraction or multiplication and division, so they cancel each other out and we're just left with $a$a!

 

Worked example

Question 12

Simplify $\left(2\sqrt{6}\right)^2$(26)2.

Think: $\left(ab\right)^2$(ab)2 = $a^2b^2$a2b2

Do:

$\left(2\sqrt{6}\right)^2$(26)2 $=$= $2^2\times\left(\sqrt{6}\right)^2$22×(6)2
  $=$= $4\times6$4×6
  $=$= $24$24

 

Practice questions

Question 13

Simplify $\sqrt[3]{9^3x^{18}y^{12}}$393x18y12.

Question 14

Simplify $\sqrt{150}$150.

Outcomes

CC.2.1.HS.F.1

Apply and extend the properties of exponents to solve problems with rational exponents

CC.2.1.HS.F.2

Apply properties of rational and irrational numbers to solve real world or mathematical problems.

A1.1.1.1.2

Simplify square roots (e.g., √24 = 2√6).

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