Roots
You want to put new flooring in a square room of $36$ m² and you need to know how long each wall is. You look for a number that, multiplied by itself, gives 36: it's 6. You've just calculated a square root. Roots undo powers, just like subtraction undoes addition.
What a root is
The square root has a very clear picture: it's the side of a square whose area you know. And the cube root is the edge of a cube whose volume you know. Try areas that aren't perfect squares:
Work out the value of the following exact roots mentally. Remember to pay attention to the index of the root to know which number multiplied by itself (two, three or…
Work out the value of the following exact roots mentally. Remember to pay attention to the index of the root to know which number multiplied by itself (two, three or more times) gives the radicand.
- a) $\sqrt{49}$
- b) $\sqrt[3]{-8}$
- c) $\sqrt[4]{16}$
- d) $\sqrt{100}$
- e) $\sqrt[3]{1000}$
- f) $\sqrt[5]{-1}$
Powers and roots are inverse operations, just like addition and subtraction. Express the following equalities in root form, following the example: Example: As…
Powers and roots are inverse operations, just like addition and subtraction. Express the following equalities in root form, following the example:
Example: As $5^2 = 25$, then $\sqrt{25} = 5$
- a) $9^2 = 81$
- b) $6^3 = 216$
- c) $(-4)^3 = -64$
- d) $10^4 = 10000$
- e) $2^5 = 32$
We want to lay cork flooring for a gym in a perfectly square room. We know that the total surface area (the area) of the room is $144 \text{ m}^2$. Calculate the length…
We want to lay cork flooring for a gym in a perfectly square room. We know that the total surface area (the area) of the room is $144 \text{ m}^2$.
Calculate the length of each side of the room. Use the following diagram to help you visualise it.
How many solutions? The sign of a root
$3^2 = 9$, but also $(-3)^2 = 9$. So 9 has two square roots: $+3$ and $-3$. On the other hand, $(-2)^3 = -8$ and no other number cubed gives $-8$: the cube root has only one solution.
| Index | Positive radicand | Negative radicand |
|---|---|---|
| Even (2, 4, …) | $\sqrt{16} = 4$; the equation $x^2 = 16$ has solutions $\pm 4$ | Doesn't exist in the reals: $\sqrt{-16}$ |
| Odd (3, 5, …) | One positive root: $\sqrt[3]{8} = 2$ | One negative root: $\sqrt[3]{-8} = -2$ |
With radicand zero, $\sqrt[n]{0} = 0$ and the equation $x^n = 0$ has a single solution: $x = 0$.
Work out the numerical value of the following exact roots. If the root has two possible solutions, state them. If it has no solution in the set of real numbers, explain…
Work out the numerical value of the following exact roots. If the root has two possible solutions, state them. If it has no solution in the set of real numbers, explain why:
- a) $\sqrt[3]{-64}$
- b) $\sqrt[4]{81}$
- c) $\sqrt[6]{-1}$
- d) $\sqrt{\frac{121}{144}}$
Solve the following simple equations where the unknown is related to a square root or a square. Remember that the inverse operation of a root is a power. a)…
Solve the following simple equations where the unknown is related to a square root or a square. Remember that the inverse operation of a root is a power.
- a) $\sqrt{x} = 9$
- b) $x^2 = 169$ (State the two possible solutions)
- c) $\sqrt{x + 1} = 5$
Inexact roots
$\sqrt{50}$ is not a whole number, because 50 is not a perfect square. But you can bound it without a calculator: $49 < 50 < 64$, so $7 < \sqrt{50} < 8$. And since 50 is very close to 49, $\sqrt{50}$ will be very close to 7 (in fact, $\sqrt{50} \approx 7.07$).
The square root of a positive integer that is not a perfect square is irrational: it has infinitely many decimal digits with no repeating period. Don't confuse "not a whole number" with "irrational": $\sqrt{0.25} = 0.5$ is rational. In problems, you either leave the root indicated (exact value) or round it (approximate value).
Not all roots give an exact integer. Without using a calculator, state between which two consecutive integers the following roots lie. Example: $\sqrt{10}$ lies between…
Not all roots give an exact integer. Without using a calculator, state between which two consecutive integers the following roots lie.
Example: $\sqrt{10}$ lies between 3 and 4, because $3^2=9$ and $4^2=16$.
- a) $\sqrt{20}$
- b) $\sqrt{55}$
- c) $\sqrt{90}$
- d) $\sqrt[3]{9}$ (Think about cubes: $1^3=1, 2^3=8, 3^3=27...$)
In mathematics, it is very important not to confuse the properties of operations. Check whether the root of a sum is equal to the sum of the roots using the following…
In mathematics, it is very important not to confuse the properties of operations. Check whether the root of a sum is equal to the sum of the roots using the following example.
Work out separately:
-
$\sqrt{64 + 36}$ (First add the numbers inside and then find the root).
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$\sqrt{64} + \sqrt{36}$ (Calculate each root separately and then add the results).
Do they give the same result? Conclude whether it is true that $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$.
Estimate the value of $\sqrt{60}$ to one decimal place without using a calculator. To do this: 1. Find the two perfect squares closest to 60. 2. Interpolate linearly or…
Estimate the value of $\sqrt{60}$ to one decimal place without using a calculator. To do this:
-
Find the two perfect squares closest to 60.
-
Interpolate linearly or use reasoned "trial and error" to approximate the first decimal place.
-
Once you have your estimate, calculate the absolute error committed knowing that the approximate real value is $7.74$.
Roots as powers
If $\left(a^{1/2}\right)^2 = a^{1/2 \times 2} = a^1 = a$, then $a^{1/2}$ is a number that squared gives $a$: it's $\sqrt{a}$! In general:
We'll use these properties with positive bases; if the exponent is positive we can also include base zero. With negative bases, the rules don't carry over without first checking the domain and the sign: for example, $\sqrt{x^2} = |x|$.
Any root can be written as a power with a fractional exponent. The general rule is $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Convert the following expressions: 1. From root to…
Any root can be written as a power with a fractional exponent. The general rule is $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.
Convert the following expressions:
- From root to power:
a) $\sqrt{5}$
b) $\sqrt[3]{2}$
c) $\sqrt[5]{x^3}$
- From power to root:
d) $3^{\frac{1}{2}}$
e) $7^{\frac{2}{3}}$
f) $a^{\frac{1}{4}}$
Simplify the following radicals to obtain an irreducible root. To do this, remember that you can divide the index of the root and the exponent of the radicand by the…
Simplify the following radicals to obtain an irreducible root. To do this, remember that you can divide the index of the root and the exponent of the radicand by the same number (HCF).
- a) $\sqrt[6]{x^4}$
- b) $\sqrt[12]{64}$ (Hint: first factorise 64)
- c) $\sqrt[10]{a^5 \times b^{15}}$