Roots

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.

1. a) $\sqrt{49}$
2. b) $\sqrt[3]{-8}$
3. c) $\sqrt[4]{16}$
4. d) $\sqrt{100}$
5. e) $\sqrt[3]{1000}$
6. 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 $5^2 = 25$, then $\sqrt{25} = 5$

1. a) $9^2 = 81$
2. b) $6^3 = 216$
3. c) $(-4)^3 = -64$
4. d) $10^4 = 10000$
5. 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 of each side of the room. Use the following diagram to help you visualise it.

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:

1. a) $\sqrt[3]{-64}$
2. b) $\sqrt[4]{81}$
3. c) $\sqrt[6]{-1}$
4. 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.

1. a) $\sqrt{x} = 9$
2. b) $x^2 = 169$ (State the two possible solutions)
3. c) $\sqrt{x + 1} = 5$

Analyse the truth of the following mathematical statements, justifying your answer with examples or counterexamples. It is not enough to say "true" or "false", you must explain why based on the definition of an $n$th root:

1. a) The square root of a positive number has two solutions, whereas the cube root of a positive number has only one.
2. b) There is no real number $x$ such that $\sqrt[4]{x} = -2$.
3. c) The expression $\sqrt[n]{a}$ exists for any real number $a$, provided that $n$ is an odd number.
4. d) If we raise a negative number to an even power and then calculate the root with the same index, we get the original number (example: $\sqrt[4]{(-3)^4} = -3$).

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:

1. $\sqrt{64 + 36}$ (First add the numbers inside and then find the root).

2. $\sqrt{64} + \sqrt{36}$ (Calculate each root separately and then add the results).

Do they give the same result? Conclude whether it is true that $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$.

Work out the exact value of the following square roots mentally or by prime factorisation. If the result is a fraction or a decimal, express it correctly.

1. a) $\sqrt{121}$
2. b) $\sqrt{400}$
3. c) $\sqrt{0.36}$
4. d) $\sqrt{\frac{64}{225}}$

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 $4^2=16$.

1. a) $\sqrt{20}$
2. b) $\sqrt{55}$
3. c) $\sqrt{90}$
4. d) $\sqrt[3]{9}$ (Think about cubes: $1^3=1, 2^3=8, 3^3=27...$)

Look at the following number line where three approximate points (A, B and C) have been marked.

Without using a calculator, match each of the following radicals with the point (A, B or C) that best represents its value, giving reasons and estimating between which integers they lie:

1. $\sqrt{5}$

2. $\sqrt[3]{2}$

3. $\sqrt{10}$

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 use reasoned "trial and error" to approximate the first decimal place.

3. Once you have your estimate, calculate the absolute error committed knowing that the approximate real value is $7.74$.

Sometimes it is necessary to insert factors into a root to compare numbers or perform certain operations. Insert the factors into the radical and simplify the result:

1. a) $3 \times \sqrt{2}$
2. b) $2 \times \sqrt[3]{5}$
3. c) $\frac{1}{2} \times \sqrt{20}$

Without using a calculator to find the decimal value, order the following numbers from smallest to largest. You must transform the numbers in order to compare them (for example, by inserting factors into the root or raising them all to a common power to eliminate the roots):

$$ A = \sqrt{10} \quad , \quad B = \sqrt[3]{30} \quad , \quad C = 3 $$

Explain step by step the method you used to compare them.

To represent irrational numbers exactly on the number line, we use Pythagoras' theorem. We know that the hypotenuse of a right-angled triangle with legs of length 1 is $\sqrt{1^2+1^2} = \sqrt{2}$. Following this logic:

Explain in detail, step by step, how you would represent the number $\sqrt{29}$ on the number line using a ruler and compasses. You must state which integers you would use as measurements for the legs of the auxiliary right-angled triangle. Use the following visual diagram to support your reasoning.

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 power:

a) $\sqrt{5}$

b) $\sqrt[3]{2}$

c) $\sqrt[5]{x^3}$

1. 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 same number (HCF).

1. a) $\sqrt[6]{x^4}$
2. b) $\sqrt[12]{64}$ (Hint: first factorise 64)
3. c) $\sqrt[10]{a^5 \times b^{15}}$

We want to store a very long metal rod inside a box shaped as a rectangular prism (a shoe box). The dimensions of the box are $30 \text{ cm}$ long, $40 \text{ cm}$ wide and $120 \text{ cm}$ high.

Calculate the exact maximum length that the rod can have to fit inside the box without bending (placing it along the space diagonal). Express the result first as a root and then calculate its exact value.

We have two spherical water tanks. We know that the volume of the large tank is 27 times the volume of the small tank.

If the radius of the small tank is $r = 50 \text{ cm}$, what is the radius of the large tank ($R$)?

Note: Remember that the volume of a sphere depends on the cube of the radius ($V \propto r^3$). You do not need the exact formula for the volume; reason using the relationship between volumes and the cube root.