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Square and cube roots

Undoing a power, and why a square root question has two answers.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

In this lesson you evaluate square and cube roots and solve equations like $x^2 = 81$ and $x^3 = -27$. A root undoes a power, which means a square root question has two answers, since $9$ and $-9$ both square to $81$, while a cube root has one. The reason is worth having: squaring destroys the sign and cubing does not.

2. What you already know

You know that $5^2$ means $5 \times 5 = 25$ and that $5^3$ means $5 \times 5 \times 5 = 125$. You have met the square numbers $1, 4, 9, 16, 25, \dots$ as the areas of squares with whole-number sides, and you can find the volume of a cube by multiplying its edge by itself three times. You also know how to multiply negative numbers: a negative times a negative is positive. In the last lessons you solved equations by undoing operations: subtracting undoes adding, and dividing undoes multiplying. This lesson adds one more pair to that list. A square root undoes squaring, and a cube root undoes cubing.

3. Words this lesson uses

4. A root undoes a power

Squaring a number means multiplying it by itself. A square root runs that backward: it asks which number, times itself, gives this one. Since $9 \times 9 = 81$, a square root of $81$ is $9$.

Cubing a number means using it three times as a factor. A cube root runs that backward: since $4 \times 4 \times 4 = 64$, the cube root of $64$ is $4$, written $\sqrt[3]{64} = 4$.

Here is the surprise. Squaring throws away the sign. $9^2 = 81$, and also $(-9)^2 = (-9) \times (-9) = 81$. So when you undo a square you cannot tell which of the two numbers you started with. That is why the equation $x^2 = 81$ has two solutions:

$$x^2 = 81 \quad\Rightarrow\quad x = 9 \text{ or } x = -9.$$

People write this as $x = \pm 9$, read "plus or minus nine". The symbol $\sqrt{81}$ on its own, though, always means the positive one, $9$. It is called the principal square root.

Cubing keeps the sign. $4^3 = 64$ but $(-4)^3 = -64$, because three negative factors make a negative product. So the equation $x^3 = 64$ has one solution, $x = 4$, and $x^3 = -64$ has one solution too, $x = -4$. You can take the cube root of a negative number; you cannot take the square root of one and get a real number, since no real number times itself is negative.

5. Squares and cubes worth knowing by heart

Roots are quick when you know the powers. These are the ones used most often in grade 8:

Read the table backward to find a root: $343$ is in the cube row under $7$, so $\sqrt[3]{343} = 7$. The squares from $11^2$ to $15^2$ are $121, 144, 169, 196, 225$, and they are worth learning too.

For a larger perfect square, two clues find the root fast. First, place the number between squares of tens: $529$ is between $20^2 = 400$ and $30^2 = 900$, so its root is in the twenties. Second, look at the last digit. Only numbers ending in $3$ or $7$ have squares ending in $9$. So try $23$: $23 \times 23 = 529$. Done.

6. Why a square root question can have two answers

Keep two questions apart, because they look alike and have different answers.

"What is $\sqrt{100}$?" The symbol asks for one number, the principal root. The answer is $10$. Writing $\sqrt{100} = -10$ is wrong.

"Solve $x^2 = 100$." This asks for every number that makes the equation true. Both $10$ and $-10$ do, so the answer is $x = 10$ or $x = -10$. Leaving out $-10$ gives only half of the answer.

A story can rule out the negative answer. If $x$ is the side of a square garden with an area of $100$ square feet, a side of $-10$ feet makes no sense, so the side is $10$ feet. The equation still has two solutions; the situation picks the one that fits. Always say which one you are keeping and why.

Some equations have no real solution at all. $x^2 = -25$ asks for a number whose square is negative. A positive times a positive is positive, and a negative times a negative is positive, so no real number works.

7. Roots of fractions and decimals

A fraction is squared by squaring its top and its bottom: $\left(\dfrac{2}{5}\right)^2 = \dfrac{4}{25}$. So its root is found the same way, one part at a time:

$$\sqrt{\dfrac{4}{25}} = \dfrac{\sqrt{4}}{\sqrt{25}} = \dfrac{2}{5}.$$

Decimals work once you write them as fractions. $0.09 = \dfrac{9}{100}$, so $\sqrt{0.09} = \dfrac{3}{10} = 0.3$. Notice that the root of a number less than $1$ is bigger than the number: $0.3$ is bigger than $0.09$. That is because multiplying by a number less than $1$ makes things smaller, so $0.3 \times 0.3$ shrinks to $0.09$.

Cube roots of fractions work the same way: $\sqrt[3]{\dfrac{8}{27}} = \dfrac{2}{3}$ because $2^3 = 8$ and $3^3 = 27$. A quick check that you have not slipped: multiply your answer back out. $\dfrac{2}{3} \times \dfrac{2}{3} \times \dfrac{2}{3} = \dfrac{8}{27}$.

8. When the root is not a whole number

Most numbers are not perfect squares. $\sqrt{2}$ is the side of a square with an area of $2$, and there is no fraction whose square is exactly $2$. Mathematicians proved this more than two thousand years ago. Numbers like this are called irrational: their decimals go on forever without repeating.

You can still pin such a root down. $1.4^2 = 1.96$ and $1.5^2 = 2.25$, so $\sqrt{2}$ is between $1.4$ and $1.5$, much closer to $1.4$. A calculator gives $1.41421356\ldots$, and the digits never settle into a pattern.

The same goes for $\sqrt{3}$, $\sqrt{5}$, $\sqrt{10}$ and cube roots like $\sqrt[3]{2}$. When an equation such as $x^2 = 10$ has an answer that is not whole, write it exactly as $x = \pm\sqrt{10}$. Round it only if the question asks for a decimal: $\sqrt{10} \approx 3.16$. The exact form is not an unfinished answer. It is the most accurate way to write the number.

9. Solving when the power is not alone

Often the square or the cube has other operations wrapped around it, as in $2x^3 + 5 = 59$. Treat $x^3$ as a block and get it alone first, undoing the operations in reverse order, just as you did with $x$ in earlier lessons. Subtract $5$ from both sides: $2x^3 = 54$. Divide both sides by $2$: $x^3 = 27$. Only now take the root: $x = 3$.

Why wait until the end? The root undoes only the power. $\sqrt[3]{2x^3 + 5}$ is not $x$ plus something simple, so taking a root too early leaves a mess. The power was the first thing done to $x$, so it is the last thing you undo.

Brackets change that order. In $(x - 1)^2 = 49$, the square was done last, so the root comes first: $x - 1 = 7$ or $x - 1 = -7$. Then add $1$ to each: $x = 8$ or $x = -6$. Both check: $7^2 = 49$ and $(-7)^2 = 49$.

10. The method, step by step, and how to check it

1. Get the power alone. Undo any adding, subtracting, multiplying or dividing around $x^2$ or $x^3$, in reverse order, doing the same to both sides.
2. Take the matching root of both sides. A square root for $x^2$, a cube root for $x^3$.
3. Decide how many answers. For a square, write $\pm$: a positive number gives two solutions, zero gives one, and a negative number gives none. For a cube, there is always exactly one real solution, with the same sign as the number.
4. Evaluate the root from the table, by the tens-and-last-digit clues, or leave it exact as $\sqrt{p}$.
5. Check every answer in the original equation.

Why the moves are allowed. If two numbers are equal and neither is negative, their square roots are equal too, because each number has just one principal root. That is what lets you take the root of both sides. The $\pm$ is there because $x$ itself may be negative.

How to check. Put each answer back into the first line and work it out. For the negative one, use brackets: $(-7)^2$ means $(-7) \times (-7) = 49$, while $-7^2$ means $-(7^2) = -49$. That bracket is where most checks go wrong.

11. In the world: laying out a baseball diamond

A Major League Baseball infield is a square with a base at each corner, and the bases are $90$ feet apart. Suppose a groundskeeper knows only that the infield square covers $8{,}100$ square feet and needs the side to mark the base paths. The side $s$ solves $s^2 = 8100$. Since $8100 = 81 \times 100$, $s = 9 \times 10 = 90$ feet. The negative solution, $-90$, is thrown out because a length cannot be negative. A Little League field has bases $60$ feet apart, so its infield covers $60^2 = 3{,}600$ square feet, less than half the area of the big-league one even though the side is two-thirds as long.

12. In the world: how long a fall lasts

When an object is dropped and air resistance is small, the distance it falls in feet is about $d = 16t^2$, where $t$ is the time in seconds. The New River Gorge Bridge in West Virginia stands $876$ feet above the river, and on Bridge Day each October people parachute from it. How long would a stone take to fall that far? Solve $16t^2 = 876$. Divide by $16$: $t^2 = 54.75$. Take the square root: $t = \sqrt{54.75} \approx 7.4$ seconds, since $7^2 = 49$ and $8^2 = 64$. Time cannot be negative, so the negative root is dropped. The fall takes a little over seven seconds, not the $55$ you might guess by forgetting the root.

13. In the world: a tank that holds 1,000 liters

One liter is $1{,}000$ cubic centimeters. A farm wants a cube-shaped water tank that holds $1{,}000$ liters, which is $1{,}000{,}000$ cubic centimeters. The edge $e$ solves $e^3 = 1{,}000{,}000$. Since $100 \times 100 \times 100 = 1{,}000{,}000$, the edge is $100$ centimeters, exactly $1$ meter, a little over $39$ inches. If the farm wanted a tank eight times as big, the edge would only double, to $2$ meters, because $\sqrt[3]{8} = 2$. Volume grows much faster than length.

14. Mistakes to watch for

Halving instead of rooting. $\sqrt{64}$ is $8$, not $32$. Ask what number times itself gives $64$, not what number plus itself.

Forgetting the negative solution. $x^2 = 36$ has two solutions, $6$ and $-6$, unless the situation rules one out.

Giving two answers to a cube. $x^3 = 125$ has only $x = 5$, because $(-5)^3 = -125$.

Writing $\sqrt{25} = \pm 5$. The symbol means the principal root, $5$. The $\pm$ belongs to solving $x^2 = 25$.

Taking the root too early. In $3x^2 = 75$, divide by $3$ before taking the square root.

15. Finding a square root by trying

1. Find $\sqrt{196}$. Place $196$ between two squares of tens.

   $10^2 = 100 < 196 < 400 = 20^2$

   So the root is a number between $10$ and $20$.

2. Look at the last digit of $196$.

   $196 \text{ ends in } 6$

   Only numbers ending in $4$ or $6$ have squares ending in $6$, since $4^2 = 16$ and $6^2 = 36$.

3. Try $14$, the first candidate.

   $14 \times 14 = 140 + 56 = 196$

   $14 \times 10 = 140$ and $14 \times 4 = 56$.

4. Write the square root.

   $\sqrt{196} = 14$

   The radical sign means the positive number whose square is $196$.

5. Note the other number with the same square.

   $(-14)^2 = 196$

   So $x^2 = 196$ would have two solutions, but $\sqrt{196}$ names only $14$.

16. The cube root of a negative number

1. Solve $x^3 = -216$. Take the cube root of both sides.

   $x = \sqrt[3]{-216}$

   The cube root undoes the cube.

2. Leave the sign aside and find the cube root of $216$.

   $6 \times 6 \times 6 = 36 \times 6 = 216$

   So $6^3 = 216$.

3. Decide the sign by cubing $-6$.

   $(-6)^3 = (-6) \times (-6) \times (-6)$

   The answer must have the sign of $-216$, so try the negative number.

4. Multiply the first two factors.

   $(-6) \times (-6) = 36$

   A negative times a negative is positive.

5. Multiply by the third factor.

   $36 \times (-6) = -216$

   The third negative factor makes the product negative again.

6. Write the solution and rule out the positive number.

   $x = -6, \quad 6^3 = 216 \ne -216$

   A cube keeps the sign, so there is exactly one real solution.

17. Getting the square alone first

1. Solve $3x^2 - 20 = 172$. Add $20$ to both sides.

   $3x^2 - 20 + 20 = 172 + 20$

   The $-20$ was done last to $x$, so it is undone first.

2. Simplify both sides.

   $3x^2 = 192$

   $-20 + 20 = 0$ and $172 + 20 = 192$.

3. Divide both sides by $3$.

   $x^2 = 192 \div 3$

   Dividing undoes multiplying by $3$.

4. Simplify the division.

   $x^2 = 64$

   Now the square stands alone.

5. Take the square root of both sides, with both signs.

   $x = \pm\sqrt{64}$

   Both a positive and a negative number could have been squared.

6. Evaluate the root.

   $x = 8 \text{ or } x = -8$

   $8 \times 8 = 64$.

7. Check $x = 8$ in the original equation.

   $3 \times 8^2 - 20 = 192 - 20 = 172$

   The left side matches the right side.

8. Check $x = -8$, using brackets.

   $3 \times (-8)^2 - 20 = 3 \times 64 - 20 = 172$

   $(-8)^2 = 64$, so the negative solution works too.

18. Your turn: solve $x^2 = 1.21$.

1. Write the decimal as a fraction.

   $1.21 = \dfrac{121}{100}$

   Two decimal places means hundredths.

2. Take the square root of the top and the bottom.

   $\sqrt{\dfrac{121}{100}} = \dfrac{11}{10} = 1.1$

   $11^2 = 121$ and $10^2 = 100$.

3. Your turn: work this step out. Its working is at the end of the packet.

   Write both solutions.

4. Your turn: work this step out. Its working is at the end of the packet.

   Check by multiplying.

19. Guided practice

Why does $x^2 = 16$ have two solutions?

• Both $4$ and $-4$ square to $16$
• Every square has only one possible root
• The exponent creates a second number by addition
• Half of $16$ is a second answer

20. Guided practice

Complete the worked solution of $3x^2 - 30 = 402$.

1. Add $30$ to both sides.

   $3x^2 =$ _p

   Adding undoes the subtraction that was done last.

2. Divide both sides by $3$.

   $x^2 =$ _q

   Now the square is alone on one side.

3. Take the square root of both sides.

   $x = \pm$ _r

   The positive root and its opposite both square to the same number.

21. Guided practice

Find $\sqrt{361}$, and the other number whose square is $361$.

$\sqrt{361} =$ _a, and the other number whose square is $361$ is _b.

22. Practice

What is $\sqrt[3]{343}$?

$\sqrt[3]{343} =$ _answer

23. Practice

Solve $x^2 = 25$. Give the positive solution and the negative solution.

The positive solution is $x =$ _p and the negative solution is $x =$ _q.

24. Practice

Solve $5x^3 = 1080$. First write what $x^3$ equals, then $x$.

$x^3 =$ _p, so $x =$ _q.

25. Somewhere new

A cube-shaped shipping box holds exactly $729$ one-inch cubes, with no gaps and nothing left over. How long is one edge, in inches, and how many square inches of cardboard cover one face?

The edge is _e inches, and one face covers _f square inches.

26. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

27. Test question

Solve $5x^2 + 20 = 425$. Give the positive solution and the negative solution.

The positive solution is $x =$ _p and the negative solution is $x =$ _q.

28. What you can do now

You can evaluate roots and solve equations with squares and cubes. Without looking: how many solutions does $x^2 = 81$ have, and how many does $x^3 = -27$ have?

Working for the steps left to you

18. Your turn: solve $x^2 = 1.21$., step 3

18. Your turn: solve $x^2 = 1.21$., step 4