In this chapter, you will revise work you have done on squares, cubes, square roots and cube roots. You will learn about laws of exponents that will enable you to do calculations using numbers written in exponential form.

Very large numbers are written in scientific notation. Scientific notation is a convenient way of writing very large numbers as a product of a number between 1 and 10 and a power of 10.

## Revision

### Exponential notation

1. Calculate.
1. $$2 \times 2 \times 2$$
2. $$2 \times 2 \times 2 \times 2 \times 2 \times 2$$
3. $$3 \times 3 \times 3$$
4. $$3 \times 3 \times 3 \times 3 \times 3 \times 3$$

Instead of writing $$3 \times 3 \times 3 \times 3 \times 3 \times 3$$ we can write $$3^6$$.

We read this as "3 to the power of 6". The number 3 is the base, and 6 is the exponent.

When we write $$3 \times 3 \times 3 \times 3 \times 3 \times 3$$ as $$3^6$$, we are using exponential notation.

1. Write each of the following in exponential form:
1. $$2 \times 2 \times 2$$
2. $$2 \times 2 \times 2 \times 2 \times 2 \times 2$$
3. $$3 \times 3 \times 3$$
4. $$3 \times 3 \times 3 \times 3 \times 3 \times 3$$
2. Calculate.
1. $$5^2$$
2. $$2^5$$
3. $$10^2$$
4. $$15^2$$
5. $$3^4$$
6. $$4^3$$
7. $$2^3$$
8. $$3^2$$

### Squares

To square a number is to multiply it by itself. The square of 8 is 64 because $$8 \times 8$$ equals 64.

We write $$8 \times 8$$ as $$8^2$$ in exponential form.

We read $$8^2$$ as eight squared.

1. Complete the table.
 Number Square the number Exponential form Square (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7 (h) 8 $$8 \times 8$$ $$8^2$$ 64 (i) 9 (j) 10 (k) 11 (l) 12
2. Calculate the following:
1. $$3^2 \times 4^2$$
2. $$2^2 \times 3^2$$
3. $$2^2 \times 5^2$$
4. $$2^2 \times 4^2$$
3. Complete the following statements to make them true:
1. $$3^2 \times 4^2 = \text{______}^2$$
2. $$2^2 \times 3^2 = \text{______}^2$$
3. $$2^2 \times 5^2 = \text{______}^2$$
4. $$2^2 \times 4^2 = \text{______}^2$$

### Cubes

To cube a number is to multiply it by itself and then by itself again. The cube of 3 is 27 because 3 \times 3 \times 3 equals 27.

We write $$3 \times 3 \times 3$$ as $$3^3$$ in exponential form.

We read $$3^3$$ as three cubed.

1. Complete the table.
 Number Cube the number Exponential form Cube (a) 1 (b) 2 (c) 3 $$3 \times 3 \times 3$$ $$3^3$$ 27 (d) 4 (e) 5 (f) 6 (g) 7 (h) 8 (i) 9 (j) 10
2. Calculate the following:
1. $$2^3 \times 3^3$$
2. $$2^3 \times 5^3$$
3. $$2^3 \times 4^3$$
4. $$1^3 \times 9^3$$
3. Which of the following statements are true? If a statement is false, rewrite it as a true statement.
1. $$2^3 \times 3^3 = 6^3$$
2. $$2^3 \times 5^3 = 7^3$$
3. $$2^3 \times 4^3 = 8^3$$
4. $$1^3 \times 9^3 = 10^3$$

### Square and cube roots

To find the square root of a number we ask the question: Which number was multiplied by itself to get a square?

The square root of 16 is 4 because $$4 \times 4 = 16$$.

The question: Which number was multiplied by itself to get 16? is written mathematically as $$\sqrt{16}$$.

The answer to this question is written as $$\sqrt{16} = 4$$.

1. Complete the table.
 Number Square of the number Square root of the square of the number Reason (a) 1 (b) 2 (c) 3 (d) 4 16 4 $$4 \times 4 = 16$$ (e) 5 (f) 6 (g) 7 (h) 8 (i) 9 (j) 10 (k) 11 (l) 12
2. Calculate the following. Justify your answer.
1. $$\sqrt{144}$$
2. $$\sqrt{100}$$
3. $$\sqrt{81}$$
4. $$\sqrt{64}$$

To find the cube root of a number we ask the question: Which number was multiplied by itself and again by itself to get a cube?

The cube root of 64 is 4 because $$4 \times 4 \times 4 = 64$$.

The question: Which number was multiplied by itself and again by itself (or cubed) to get 64? is written mathematically as $$\sqrt{64}$$.

The answer to this question is written as $$\sqrt{64} = 4$$.

1. Complete the table.
 Number Cube of the number Cube root of the cube of the number Reason (a) 1 (b) 2 (c) 3 (d) 4 64 4 $$4 \times 4 \times 4 = 64$$ (e) 5 (f) 6 (g) 7 (h) 8 (i) 9 (j) 10
2. Calculate the following and give reasons for your answers:
1. $$\sqrt{216}$$
2. $$\sqrt{8}$$
3. $$\sqrt{125}$$
4. $$\sqrt{27}$$
5. $$\sqrt{64}$$
6. $$\sqrt{1000}$$

## Working with integers

### Representing integers in exponential form

1. Calculate the following, without using a calculator:
1. $$-2 \times -2 \times -2$$
2. $$-2 \times -2 \times -2 \times -2$$
3. $$-5 \times -5$$
4. $$-5 \times -5 \times -5$$
5. $$-1 \times -1 \times -1 \times -1$$
6. $$-1 \times -1 \times -1$$
2. Calculate the following:
1. $$-2^2$$
2. $$(-2)^2$$
3. $$(-5)^2$$
4. $$-5^3$$
3. Use your calculator to calculate the answers to question 2.
1. Are your answers to question 2(a) and (b) different or the same as those of the calculator?
2. If your answers are different to those of the calculator, try to explain how the calculator did the calculations differently from you.

The calculator "understands" $${\bf-5^2}$$ and $${\bf(-5)^2}$$ as two different numbers.

I understands $${\bf-5^2}$$ as $${\bf-5 \times 5 = -25}$$ and $${\bf(-5)^2}$$ as $${\bf-5 \times -5 = 25}$$

1. Write the following in exponential form:
1. $$-2 \times -2 \times -2$$
2. $$-2 \times -2 \times -2 \times -2$$
3. $$-5 \times -5$$
4. $$-5 \times -5 \times -5$$
5. $$-1 \times -1 \times -1 \times -1$$
6. $$-1 \times -1 \times -1$$
2. Calculate the following:
1. $$(-3)^2$$
2. $$(-3)^3$$
3. $$(-2)^4$$
4. $$(-2)^6$$
5. $$(-2)^5$$
6. $$(-3)^4$$
3. Say whether the sign of the answer is negative or positive. Explain why.
1. $$(-3)^6$$
2. $$(-5)^{11}$$
3. $$(-4)^{20}$$
4. $$(-7)^5$$
4. Say whether the following statements are true or false. If a statement is false rewrite it as a correct statement.
1. $$(-3)^2 = -9$$
2. $$-3^2 = 9$$
3. $$(-5^2) = 5^2$$
4. $$(-1)^3 = -1^3$$
5. $$(-6)^3 = -18$$
6. $$(-2)^6 = 2^6$$

## Laws of exponents

### Product of powers

1. A product of 2s is given below. Describe it using exponential notation, that is, write it as a power of 2.

$$2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$$

2. Express each of the following as a product of the powers of 2, as indicated by the brackets.
1. $$(2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2)$$
2. $$(2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2)$$
3. $$(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)$$
4. $$(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)$$
5. $$(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2)$$
3. Complete the following statements so that they are true. You may want to refer to your answers to question 2 (a) to (e) to help you.
1. $$2^3 \times \text{______} = 2^{12}$$
2. $$2^5 \times \text{______} \times 2^2 = 2^{12}$$
3. $$2^2 \times 2^2 \times 2^2 \times 2^2 \times 2^2 \times 2^2 = \text{______}$$
4. $$2^8 \times \text{______} = 2^{12}$$
5. $$2^3 \times 2^3 \times 2^3 \times \text{______} = 2^{12}$$
6. $$2^6 \times \text{______} = 2^{12}$$
7. $$2^2 \times 2^{10} \times = \text{______}$$

Suppose we are asked to simplify: $$3^2 \times 3^4$$.

\begin{align} \text{The solution is: }3^3 \times 3^4 & = 9 \times 81 \\ &= 729 \\ &= 3^6 \end{align}

The base (3) is a repeated factor. The exponents (2 and 4) tell us the number of times each factor is repeated.

We can explain this solution in the following manner:

\begin{align} 3^2 \times 3^4 = \underbrace{3 \times 3}_\text{2 factors} \times \underbrace{3 \times 3 \times 3 \times 3}_\text{4 factors} = \underbrace{3 \times 3 \times 3 \times 3 \times 3 \times 3}_\text{6 factors} = 3^6 \end{align}

1. Complete the table.
 Product of powers Repeated factor Total number of times the factor is repeated Simplified form (a) $$2^7 \times 2^3$$ (b) $$5^2 \times 5^4$$ (c) $$4^1 \times 4^5$$ (d) $$6^3 \times 6^2$$ (e) $$2^8 \times 2^2$$ (f) $$5^3 \times 5^3$$ (g) $$4^2 \times 4^4$$ (h) $$2^1 \times 2^9$$

When you multiply two or more powers that have the same base, the answer has the same base, but its exponent is equal to the sum of the exponents of the numbers you are multiplying.

We can express this symbolically as $$a^m \times a^n = a^{m+n}$$, where m and n are natural numbers and a is not zero.

1. What is wrong with these statements? Correct each one.
1. $$2^3 \times 2^4 = 2^{12}$$
2. $$10 \times 10^2 \times 10^3 \times = 10^{1 \times 2 \times 3} = 10^6$$
3. $$3^2 \times 3^2 = 3^6$$
4. $$5^3 \times 5^2 = 15 \times 10$$
2. Express each of the following numbers as a single power of 10.

Example: 1 000 000 as a power of 10 is $$10^6$$.

1. $$100$$
2. $$10 00$$
3. $$100 000$$
4. $$10^2 \times 10^3 \times 10^4$$
5. $$100 \times 1000 \times 10000$$
6. $$1000000000$$
3. Write each of the following products in exponential form:
1. $$x \times x \times x \times x \times x \times x \times x \times x \times x$$
2. $$(x \times x) \times (x \times x \times x) \times (x \times x \times x \times x)$$
3. $$(x \times x \times x \times x) \times (x \times x) \times (x \times x) \times x$$
4. $$(x \times x \times x \times x \times x \times x) \times (x \times x \times x)$$
5. $$(x \times x \times x) \times (y \times y \times y)$$
6. $$(a \times a) \times (b \times b)$$
4. Complete the table.
 Product of powers Repeated factor Total number of times the factor is repeated Simplified form (a) $$x^7 \times x^3$$ (b) $$x^2 \times x^4$$ (c) $$x^1 \times x^5$$ (d) $$x^3 \times x^2$$ (e) $$x^8 \times x^2$$ (f) $$x^3 \times x^3$$ (g) $$x^1 \times x^9$$

### Raising a power to a power

1. Complete the table of powers of 2.
 x 1 2 3 4 5 6 7 8 9 10 11 $$2^x$$ 2 4 $$2^1$$ $$2^2$$ $$2^3$$
 x 12 13 14 15 16 17 18 $$2^x$$
2. Complete the table of powers of 3.
 x 1 2 3 4 5 6 7 8 9 $$3^x$$ $$3^1$$ $$3^2$$ $$3^3$$
 $$x$$ 10 11 12 13 14 $$3^x$$
3. Complete the table. You can read the values from the tables you made in questions 1 and 2.
 Product of powers Repeated factor Power of powernotation Total number of repetitions Simplified form Value $$2^4 \times 2^4 \times 2^4$$ 2 $$(2^4)^3$$ 12 $$2^{12}$$ 4 096 $$3^2 \times 3^2 \times 3^2 \times 3^2$$ $$2^3 \times 2^3 \times 2^3 \times 2^3 \times 2^3$$ $$3^4 \times 3^4 \times 3^4$$ $$2^6 \times 2^6 \times 2^6$$
4. Use your table of powers of 2 to find the answers for the following:
1. $$2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = \text{______} = \text{______}$$
2. $$(2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = \text{______} = \text{______}$$
3. $$16^3 = \text{______} = \text{_______} = \text{______}$$
5. Use your table of powers of 2 to find the answers for the following:
1. Is $$16^3 = 2^{12}$$?
2. Is $$2^4 \times 2^4 \times 2^4 = 2^{12}$$?
3. Is $$2^4 \times 2^3 = 2^{12}$$?
4. Is $$(2^4)^3 = 2^4 \times 2^4 \times 2^4$$?
5. Is $$(2^4)^3 = 2^{12}$$?
6. Is $$(2^4)^3 = 2^{4+3}$$?
7. Is $$(2^4)^3 = 2^{4 \times 3}$$?
8. Is $$(2^2)^5 = 2^{2+5}$$?
1. Express $$8^5$$ as a power of 2. It may help to first express 8 as a power of 2.
2. Can $$(2^3) \times (2^3) \times (2^3) \times (2^3) \times (2^3)$$ be expressed as $$(2^3)^5$$?
3. Is $$(2^3)^5 = 2^{3+5}$$ or is $$(2^3)^5 = 2^{3 \times 5}$$?
1. Express $$4^3$$ as a power of 2.
2. Calculate $$2^2 \times 2^2 \times 2^2$$ and express your answer as a single power of 2.
3. Can $$(2^2) \times (2^2) \times (2^2)$$ be expressed as $$(2^2)^3$$?
4. Is $$(2^2)^3 = 2^{2+3}$$ or is $$(2^2)^3 = 2^{2 \times 3}$$?
6. Simplify the following.

Example: $$(10^2)^2 = 10^2 \times 10^2 = 10^{2+2}=10^4 = 10000$$

1. $$(3^2)^2$$
2. $$(4^3)^2$$
3. $$(2^4)^2$$
4. $$(9^2)^2$$
5. $$(3^3)^3$$
6. $$(4^3)^3$$
7. $$(5^4)^3$$
8. $$(9^2)^3$$

$$(a^m)^n = a^{m \times n}$$, where m and n are natural numbers and a is not equal to zero.

1. Simplify.
1. $$(5^4)^{10}$$
2. $$(10^4)^5$$
3. $$(6^4)^4$$
2. Write $$5^{12}$$ as a power of powers of 5 in two different ways.

To simplify $$(x^2)^5$$ we can write it out as a product of powers or we can use a shortcut.

\begin{align} (x^2)^5 &= (x^2) \times (x^2) \times (x^2) \times (x^2) \times (x^2) \\ \ &= \underbrace{x \times x} \times \underbrace{x \times x} \times \underbrace{x \times x} \times \underbrace{x \times x} \times \underbrace{x \times x} = x^{10} \\ & 2 \times 5 \text{ factors} = 10 \text{ factors} \end{align}

1. Complete the table.
 Expression Write as a product of the powers and then simplify Use the rule $${\bf(a^m)^n}$$ to simplify (a) $$(a^4)^5$$ $a^4 \times a^4 \times a^4 \times a^4 \times a^4 \\ a^{4 + 4 +4 + 4 + 4} = a^{20}$ $$(a^4)^5 = a^{4 \times 5} =a ^{20}$$ (b) $$(b^{10})^5$$ (c) $$(x^7)^3$$ (d) $s^6 \times s^6 \times s^6 \times s^6 \\ = s^{6+6+6+6} \\ = s^{24}$ (e) $$y^{3 \times 7} = y^{21}$$

### Power of a product

1. Complete the table. You may use your calculator when you are not sure of a value.
 $$x$$ 1 2 3 4 5 (a) $$2^x$$ $$2^1 = 2$$ (b) $$3^x$$ $$3^2 = 9$$ (c) $$6^x$$ $$6^3 = 216$$
2. Use the table in question 1 to answer the questions below. Are these statements true or false? If a statement is false rewrite it as a correct statement.
1. $$6^2 = 2^2 \times 3^2$$
2. $$6^3 = 2^3 \times 3^3$$
3. $$6^5 = 2^5 \times 3^5$$
4. $$6^8 = 2^4 \times 3^4$$
3. Complete the table.
 Expression The bases of the expression are factors of . . . Equivalent expression (a) $$2^6 \times 2^5$$ 10 $$10^6$$ (b) $$3^2 \times 4^2$$ (c) $$4^2 \times 2^2$$ (d) $$56^5$$ (e) $$30^3$$ (f) $$3^5 \times x^5$$ $$3x$$ $$(3x)^5$$ (g) $$7^2 \times z^2$$ (h) $$4^3 \times y^3$$ (i) $$(2m)^6$$ (j) $$(2m)^3$$ (k) $$2^{10} \times y^{10}$$ $$(2y)^{10}$$

$$12^2$$can be written in terms of its factors as $$(2 \times 6)^2$$ or as $$(3 \times4)^2$$

We already know that $$12^2$$ = 144.

What this tells us is that both $$(2 \times 6)^2$$ and $$(3 \times4)^2$$ also equal 144.

We write 12:

\begin{align} 12^2 &= (2 \times 6)^2 \\ &= 2^2 \times 6^2 \\ &= 4 \times 36 \\ &= 144 \end{align}

or

\begin{align} 12^2 &= (3 \times 4)^2 \\ &= 3^2 \times 4^2 \\ &= 9 \times 16 \\ &= 144 \end{align}

A product raised to a power is the product of the factors each raised to the same power.

Using symbols, we write $$(a\times b)^m = a^m \times b^m$$, where m is a natural number and a and b are not equal to zero

1. Write each of the following expressions as an expression with one base:

Example: $$3^{10} \times 2^{10} = (3 \times 2)^{10} = 6^{10}$$

1. $$3^2 \times 5^2$$
2. $$5^3 \times 2^3$$
3. $$7^4 \times 4^4$$
4. $$2^3 \times 6^3$$
5. $$4^4 \times 2^4$$
6. $$5^2 \times 7^2$$
2. Write the following as a product of powers:

Example:$$(3x)^3 = 3^3 \times x^3 = 27x^3$$

1. $$6^3$$
2. $$15^2$$
3. $$21^4$$
4. $$6^5$$
5. $$18^2$$
6. $$(st)^7$$
7. $$(ab)^3$$
8. $$(2x)^2$$
9. $$(3y)^5$$
10. $$(3c)^2$$
11. $$(gh)^4$$
12. $$(4x)^3$$
3. Simplify the following expressions:

Example: $$3^2 \times m^2 = 9 \times m^2 = 9m^2$$

1. $$3^5 \times b^5$$
2. $$2^6 \times y^6$$
3. $$x^2 \times y^2$$
4. $$10^4 \times x^4$$
5. $$3^3 \times x^3$$
6. $$5^2 \times t^2$$
7. $$6^3 \times m^7$$
8. $$12^2 \times a^2$$
9. $$n^3 \times p^9$$

### A quotient of powers

Consider the following table:

 $$x$$ 1 2 3 4 5 6 $$2^x$$ 2 4 8 16 32 64 $$3^x$$ 3 9 27 81 243 729 $$5^x$$ 5 25 125 625 3 125 15 625

Answer questions 1 to 4 by referring to the table when you need to.

1. Give the value of each of the following:
1. $$3^4$$
2. $$2^5$$
3. $$5^6$$
1. Calculate $$3^6 \div 3^3$$ (Read the values of $$3^6$$ and $$3^3$$ from the table and then divide. You may use a calculator where necessary.)

To calculate $$4^{5-3}$$ we first do the calculation in the exponent, that is, we subtract 3 from 5. Then we can calculate $$4^2$$ as $$4 \times 4 = 16$$.

2. Calculate $$3^{6-3}$$
3. Is $$3^6 \div 3^3$$ equal to $$3^3$$? Explain.
1. Calculate the value of $$2^{6-2}$$
2. Calculate the value of $$2^6 \div 2^2$$
3. Calculate the value of $$2^{6\div 2}$$
4. Read from the table the value of $$2^3$$
5. Read from the table the value of $$2^4$$
6. Which of the statements below is true? Give an explanation for your answer.

A. $$2^6 \div 2^2 = 2^{6-2} = 2^4$$

B. $$2^6 \div 2^2 = 2^{6 \div 2} = 2^3$$

2. Say which of the statements below are true and which are false. If a statement is false rewrite it as a correct statement.
1. $$5^6 \div 5^4 = 5^{6 \div 4}$$
2. $$3^{4-1} = 3^4 \div 3$$
3. $$5^6 \div 5 = 5^{6 - 1}$$
4. $$2^5 \div 2^3 = 2^2$$

$$a^m \div a^n = a^{m-n}$$

where m and n are natural numbers and m is a number greater than n and a is not zero.

1. Simplify the following. Do not use a calculator.

Example: $$3^{17} \div 3^{12} = 3^{17-12} = 3^5 = 243$$

1. $$2^{12} \div 2^{10}$$
2. $$6^{17} \div 6^{14}$$
3. $$10^{20} \div 10^{14}$$
4. $$5^{11} \div 5^{8}$$
2. Simplify:
1. $$x^{12} \div x^{10}$$
2. $$y^{17} \div y^{14}$$
3. $$t^{20} \div t^{14}$$
4. $$n^{11} \div n^{8}$$

### The power of zero

1. Simplify the following:
1. $$2^{12} \div 2^{12}$$
2. $$6^{17} \div 6^{17}$$
3. $$6^{14} \div 6^{14}$$
4. $$2^{10} \div 2^{10}$$

We define $$a^0 = 1$$

Any number raised to the power of zero is always equal to 1.

2. Simplify the following:
1. $$100^0$$
2. $$x^0$$
3. $$(100x)^0$$
4. $$(5x^3)^0$$

## Calculations

### Mixed operations

Simplify the following:

1. $$3^3 + \sqrt{-27} \times 2$$
2. $$5 \times (2+3)^2 + (-1)^0$$
3. $$3^2 \times 2^3 + 5 \times \sqrt{100}$$
4. $$\frac{ \sqrt{1000}}{\sqrt{100}} + (4-1)^2$$
5. $$\sqrt{16} \times \sqrt{16} + \sqrt{216} + 3^2 \times 10$$
6. $$4^3 \div 2^3 + \sqrt{144}$$

## Squares, cubes and roots of rational numbers

### Squaring a fraction

Squaring or cubing a fraction or a decimal fraction is no different from squaring or cubing an integer.

1. Complete the table.
 Fraction Square the fraction Value of the square of the fraction (a) $$\frac{1}{2}$$ $$\frac{1}{2} \times \frac{1}{2}$$ $$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$ (b) $$\frac{2}{3}$$ (c) $$\frac{3}{4}$$ (d) $$\frac{2}{5}$$ (e) $$\frac{3}{5}$$ (f) $$\frac{2}{6}$$ (g) $$\frac{3}{7}$$ (h) $$\frac{11}{12}$$
2. Calculate the following:
1. $$(\frac{3}{2})^2$$
2. $$(\frac{4}{5})^2$$
3. $$(\frac{7}{8})^2$$
1. Use the fact that 0,6 can be written$$\frac{6}{10}$$ to calculate $$(0,6)^2$$.
2. Use the fact that 0,8 can be written as $$\frac{8}{10}$$ to calculate $$(0,8)^2$$.

### Finding the square root of a fraction

1. Complete the table.
 Fraction Writing the fraction as a product of factors Square root (a) $$\frac{81}{121}$$ (b) $$\frac{64}{81}$$ (c) $$\frac{49}{169}$$ (d) $$\frac{100}{225}$$
2. Determine the following:
1. $$\sqrt{\frac{25}{16}}$$
2. $$\sqrt{\frac{81}{144}}$$
3. $$\sqrt{\frac{400}{900}}$$
4. $$\sqrt{\frac{36}{81}}$$
1. Use the fact that 0,01 can be written as $$\frac{1}{100}$$ to calculate $$\sqrt{0.01}$$.
2. Use the fact that 0,49 can be written as$$\frac{49}{100}$$ to calculate $$\sqrt{0.49}$$.
3. Calculate the following
1. $$\sqrt{0.09}$$
2. $$\sqrt{0.64}$$
3. $$\sqrt{1.44}$$

### Cubing a fraction

One half cubed is equal to one eighth.

We write this as $$(\frac{1}{2})^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$$

1. Calculate the following:
1. $$(\frac{2}{3})^3$$
2. $$(\frac{5}{10})^3$$
3. $$(\frac{5}{6})^3$$
4. $$(\frac{4}{5})^3$$
1. Use the fact that 0,6 can be written$$\frac{6}{10}$$ to calculate $$(0,6)^3$$.
2. Use the fact that 0,8 can be written as $$\frac{8}{10}$$ to calculate $$(0,8)^3$$.
3. Use the fact that 0,7 can be written as $$\frac{7}{10}$$ to calculate $$(0,7)^3$$.

## Scientific notation

### Very large numbers

1. Express each of the following as a single number. Do not use a calculator.

Example: $$7,56 \times 100$$ can be written as 756.

1. $$3,45 \times 100$$
2. $$3,45 \times 10$$
3. $$3,45 \times 1 000$$
4. $$2,34 \times 10^2$$
5. $$2,34 \times 10$$
6. $$2,34 \times 10^3$$
7. $$10^4 \times 10^2$$
8. $$10^0 \times 10^6$$
9. $$3,4 \times 10^5$$

We can write 136 000 000 as $$1,36 \times 10^8$$.

$$1,36 \times 10^8$$ is called the scientific notation for 136 000 000.

In scientific notation, a number is expressed in tow parts: a number between 1 and 10 multiples by a power of 10. The exponent must always be an integer.

1. Write the following numbers in scientific notation:
1. 367 000 000
2. 21 900 000
3. 600 000 000 000
4. 178
2. Write each of thefollowing numbers in the ordinary way.

For example: $$3,4 \times 10^5$$ written in the ordinary way is 340 000.

1. $$1,24 \times 10^8$$
2. $$9,2074 \times 10^4$$
3. $$1,04 \times 10^6$$
4. $$2,05 \times 10^3$$
3. The age of the universe is 15 000 000 000 years. Express the age of the universe in scientific notation.
4. The average distance from the Earth to the Sun is 149 600 000 km. Express this distance in scientific notation.

Because it is easier to multiply powers of ten without a calculator, scientific notation makes it possible to do calculations in your head.

5. Explain why the number $$24 \times 10^3$$ is not in scientific notation.
6. Calculate the following. Do not use a calculator.

Example: $$3 000 000 \times 90 000 000 = 3 \times 10^6 \times 9 \times 10^7 = 3 \times 9 \times 10^{6 + 7} \\ = 27 \times 10^{13} = 270 000 000 000 000$$

1. $$13 000 \times 150 000$$
2. $$200 \times 6 000 000$$
3. $$120 000 \times 120 000 000$$
4. $$2,5 \times 40 000 000$$
7. Use > or < to compare these numbers:
1. $$1,3 \times 10^9$$ ☐ $$2,4 \times 10^7$$
2. $$6,9 \times 10^2$$ ☐ $$4,5 \times 10^3$$
3. $$7,3 \times 10^4$$ ☐ $$7,3 \times 10^2$$
4. $$3,9 \times 10^6$$ ☐ $$3,7 \times 10^7$$
1. Calculate:
1. $$11^2$$
2. $$3^2 \times 4^2$$
3. $$6^3$$
4. $$\sqrt{121}$$
5. $$(-3)^2$$
6. $$\sqrt{125}$$
2. Simplify
1. $$3^4 \times m^6$$
2. $$b^2 \times n^6$$
3. $$y^{12} \div y^5$$
4. $$(10^2)^3$$
5. $$(2w^2)^3$$
6. $$(3d^5)(2d)^3$$
3. Calculate
1. $$(\frac{2}{5})^2$$
2. $$\sqrt{\frac{9}{25}}$$
3. $$(6^4y^2)^0$$
4. $$(0.7)^2$$
4. Simplify
1. $$(2^2 + 4)^2 + \frac{6^2}{3^3}$$
2. $$\sqrt{-125} -5 \times 3^2$$
5. Write $$3 \times 10^9$$ in the ordinary way.
6. The first birds appeared on Earth about 208 000 000 years ago. Write this number in scientific notation.
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