Powers Roots And Standard Form

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Notes

Powers & Roots

**Powers (indices)** show repeated multiplication: $e.g., 6^{3} = 6 \times 6 \times 6$ .
Any non-zero number to the power of 0 equals **1**: e.g., 3⁰ $= 1$ .
Any number to the power of 1 equals itself: e.g., 3¹ $= 3$ .
**Square roots** are the reverse of squaring; every positive number has two square roots (positive and negative).
The symbol √ denotes the **positive square root** only: $e.g., \sqrt{25} = 5$ .
**Cube roots** are the reverse of cubing; each number has only one cube root: $e.g., ^{3}\sqrt{125} = 5$ .
For **nth roots**, if n is even there are two roots (positive and negative); if n is odd there is one.
The **reciprocal** of a number is 1 divided by that number; a number and its reciprocal multiply to 1.

Laws of Indices

**a¹ = a** – anything to the power 1 is itself.
**a⁰ = 1** – anything (non-zero) to the power 0 is 1.
**aᵐ × aⁿ = aᵐ⁺ⁿ** – to multiply indices with the same base, add the powers.
**aᵐ ÷ aⁿ = aᵐ⁻ⁿ** – to divide indices with the same base, subtract the powers.
**(aᵐ)ⁿ = aᵐⁿ** – to raise a power to another power, multiply the powers.
**(ab)ⁿ = aⁿbⁿ** – to raise a product to a power, apply the power to each factor.
**(a/b)ⁿ = aⁿ / bⁿ** – to raise a fraction to a power, apply the power to numerator and denominator.
**a⁻ⁿ = 1/aⁿ** – a negative power means the reciprocal of the positive power.

Converting to & from Standard Form

Standard form is **a × 10ⁿ** where **1 $\le a <$ 10** and n is an integer.
For large numbers $(\ge 10), n$ is **positive**: count how many places the decimal moves left.
For small numbers $(0 <$ number $< 1), n$ is **negative**: count how many places the decimal moves right.
Example: $32 400 = 3.24 \times$ 10⁴ (decimal moves 4 places left).
Example: $0.0000324 = 3.24 \times$ 10⁻⁵ (decimal moves 5 places right).
To convert from standard form to ordinary number, move the decimal point n places (right if $n>0$ , left if $n<0)$ .

Operations with Standard Form

Use your calculator for calculations with standard form (calculator paper only).
Enter numbers using the **×10ˣ** button and brackets: e.g., (3×10⁸)×(2×10⁻³).
If the calculator output is not in standard form, rewrite it: $e.g., 37 500 000 =$ 3.75×10⁷.
Alternatively, rewrite a in standard form and apply index laws: e.g., 243×10²⁰ = (2.43×10²)×10²⁰ $= 2.43\times 10^{2}^{2}$ .

Worked Examples from Past Papers

Write 0.0018 in standard form: **1.8 × 10⁻³**.
Write 6.09×10⁸ as an ordinary number: **609 000 000**.
Find 5⁰: **1**; find 5⁻²: **1/25**.
Calculate (4.1×10⁻³)×(8.9×10⁷) = **3.649×10⁵** (using calculator).
Solve 5ˣ $\times 5^{3} =$ 5¹² → $x =$ **9** $(since x+3=12)$ .
Solve 5ᵖ ÷ 5⁸ = 5¹³ → $p =$ **21** $(since p-8=13)$ .

Powers and Roots Relationships

Converting to Standard Form

Operations with Standard Form

Practice questions

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1.Write down the value of $19^{0}$ .
Easy
- A0
- B1
- C19
- D $19^{0}$
2.The cost of building a ship was $153000000. Write 153000000 in standard form.
Easy
- A $1.53 \times 10^{8}$
- B $1.53 \times 10^{7}$
- C $15.3 \times 10^{7}$
- D $1.53 \times 10^{9}$
3.Write the number 40 in standard form.
Easy
- A $4 \times 10^{1}$
- B $40 \times 10^{0}$
- C $4.0 \times 10^{1}$
- D $0.4 \times 10^{2}$
4.Write 2020 in standard form.
Easy
- A $2.02 \times 10^{3}$
- B $2.02 \times 10^{4}$
- C $20.2 \times 10^{2}$
5.Write 15060 in standard form.
Easy
- A $1.506 \times 10^{4}$
- B $1.506 \times 10^{3}$
- C $15.06 \times 10^{3}$
- D $1.506 \times 10^{5}$
6.Write 72000 in standard form.
Easy
- A $7.2 \times 10^{4}$
- B $7.2 \times 10^{3}$
- C $72 \times 10^{3}$
- D $7.2 \times 10^{5}$
7.Write 0.0018 in standard form.
Easy
- A $1.8 \times 10^{-3}$
- B $1.8 \times 10^{-4}$
- C $18 \times 10^{-4}$
- D $1.8 \times 10^{3}$
8.Write down the value of $25^{0}$ .
Easy
- A0
- B1
- C25
- D $25^{0}$

Summary

## Powers, Roots & Standard Form ### Powers & Roots - **Powers (indices)**: small number showing repeated multiplication. e.g., 6³ = 6×6×6. - **Any non-zero number to power 0 = 1** (e.g., 3⁰ = 1). - **Any number to power 1 = itself** (e.g., 3¹ = 3). - **Square roots**: √25 = 5 (positive root). Every positive number has two square roots (±). Negative numbers have no real square root. - **Cube roots**: ³√125 = 5. Each number has one cube root. - **nth roots**: For even n, positive and negative roots exist; negative numbers have no real even root. For odd n, one real root. - **Reciprocal**: 1 divided by the number. e.g., reciprocal of 2 is ½. Written as index -1: 5⁻¹ = 1/5. ### Laws of Indices - a¹ = a - a⁰ = 1 - aᵐ × aⁿ = aᵐ⁺ⁿ - aᵐ ÷ aⁿ = aᵐ⁻ⁿ - (aᵐ)ⁿ = aᵐⁿ - (ab)ⁿ = aⁿbⁿ - (a/b)ⁿ = aⁿ/bⁿ - a⁻ⁿ = 1/aⁿ ### Standard Form - **Form**: A × 10ⁿ where 1 ≤ A < 10 and n is integer. - **Large numbers**: n positive. Move decimal left. e.g., 32400 = 3.24 × 10⁴. - **Small numbers**: n negative. Move decimal right. e.g., 0.0000324 = 3.24 × 10⁻⁵. - **Converting from standard form**: Move decimal n places (right if n>0, left if n<0). ### Operations with Standard Form - Use calculator for calculations (calculator paper). - Multiply/divide numbers and powers separately, then adjust to standard form. - Example: (1.275×10⁶) ÷ (3.4×10⁻²) = 3.75×10⁷.

Key terms

Power (Index): A small number showing how many times a base is multiplied by itself.
Base: The big number that is raised to a power.
Square root: A value that when squared equals the original number. Notation: √.
Cube root: A value that when cubed equals the original number. Notation: ³√.
Reciprocal: 1 divided by a number. The reciprocal of x is 1/x.
Standard form: A number written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
Laws of indices: Rules for simplifying expressions with powers, e.g., aᵐ × aⁿ = aᵐ⁺ⁿ.

Slides

Powers and Roots

Powers (indices) show repeated multiplication: $e.g., 6^{3} = 6 \times 6 \times 6$ .
Any non-zero number to the power of 0 equals 1: e.g., 3⁰ $= 1$ .
Any number to the power of 1 equals itself: e.g., 3¹ $= 3$ .
Square roots: $\sqrt{25} = 5 (positive root)$ . Every positive number has two square roots (±).
Cube roots: ∛125 $= 5$ . Each number has one cube root.

What is the value of 19⁰?

0
1
19
19¹

Reciprocals and Negative Indices

The reciprocal of a number is 1 divided by that number: e.g., reciprocal of 2 is ½.
A negative index means reciprocal: e.g., 5⁻¹ $= \frac{1}{5}$ , 5⁻² $= \frac{1}{5}^{2} = \frac{1}{25}$ .
5⁻² $= \frac{1}{25}$ .

The value of 5⁻² $is ____$ .

Laws of Indices

Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ $(e.g., 4^{3} \times 4^{2} =$ 4⁵).
Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (e.g., 7⁵ $\div 7^{2} = 7^{3})$ .
Power of a power: (aᵐ)ⁿ = aᵐⁿ.
These laws only work when bases are the same.

Simplify 5ⁿ ÷ 5⁴ = 5⁶. What is n?

Standard Form: Large Numbers

Standard form: $A \times$ 10ⁿ, where $1 \le A < 10$ and n is an integer.
For large numbers $(\ge 10), n$ is positive.
Example: $32,400 = 3.24 \times$ 10⁴ (move decimal 4 places left).
Example: $153,000,000 = 1.53 \times$ 10⁸.

Write 640,000 in standard form: ____.

Standard Form: Small Numbers

For small numbers $(0 <$ number $< 1), n$ is negative.
Example: $0.0000324 = 3.24 \times$ 10⁻⁵ (move decimal 5 places right).
Example: $0.0018 = 1.8 \times$ 10⁻³.

What is 0.0006 in standard form?

$6 \times$ 10⁻⁴
$6 \times$ 10⁻³
$6 \times$ 10⁴
$0.6 \times$ 10⁻³

Converting from Standard Form to Ordinary Number

To convert from standard form, multiply A by 10ⁿ.
If n is positive, move decimal point right n places.
If n is negative, move decimal point left n places.
Example: $6.09 \times$ 10⁸ $= 609,000,000$ .

Write $2.06 \times$ 10⁻² as an ordinary number: ____.

Operations with Standard Form

Use a calculator for operations with standard form (calculator paper).
Enter numbers using the ×10ˣ button and brackets.
If answer is not in standard form, convert it: $e.g., 37,500,000 = 3.75 \times$ 10⁷.
Alternatively, rewrite A in standard form and use index laws.

Calculate $(4.1 \times$ 10⁻³) $\times (8.9 \times$ 10⁷) and give answer in standard form.

$3.649 \times$ 10⁵
$3.649 \times$ 10⁴
$3.649 \times$ 10⁻²¹
$3.649 \times$ 10⁻⁴

Which of these are true?

3⁰ $= 0$
5⁻² $= \frac{1}{25}$
$\sqrt{25} = \pm 5$
$0.0018 = 1.8 \times$ 10⁻³

Summary

Powers: a⁰=1, a¹=a, negative indices give reciprocals.
Index laws: multiply → add powers, divide → subtract powers.
Standard form: $A \times$ 10ⁿ with $1\le A<10$ .
Large numbers: positive n; small numbers: negative n.
Use calculator for operations, then convert to standard form.

Flashcards

What is the value of any non-zero number raised to the power of 0?

Any non-zero number to the power of 0 is equal to 1. For example, $3^{0} = 1$ .

What is the value of any number raised to the power of 1?

Any number to the power of 1 is equal to itself. For example, $3^{1} = 3$ .

What are the two square roots of 25?

The two square roots of 25 are 5 and -5, because $5^{2} = 25$ and $(-5)^{2} = 25$ .

What does the notation $\sqrt{25}$ represent?

The notation $\sqrt{25}$ refers to the positive square root, which is 5.

What is the cube root of 125?

The cube root of 125 is 5, because $5^{3} = 125$ .

How many real cube roots does a number have?

Every positive and negative number has exactly one real cube root.

What is the reciprocal of a number?

The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 2 is 1/2.

How do you write the reciprocal of 5 using an index?

The reciprocal of 5 is written as $5^(-1)$ .

What is the law of indices for multiplying powers with the same base?

To multiply indices with the same base, add their powers: $a^{m} \times a^{n} = a^(m+n)$ .

What is the law of indices for dividing powers with the same base?

To divide indices with the same base, subtract their powers: $a^{m} \div a^{n} = a^(m-n)$ .

What is the law of indices for raising a power to another power?

To raise a power to another power, multiply the powers: $(a^{m})^{n} = a^(mn)$ .

What does a negative index represent?

A negative index represents the reciprocal: $a^(-n) = 1/(a^{n})$ . For example, $2^(-4) = \frac{1}{16}$ .

Write 153000000 in standard form.

$153000000 = 1.53 \times 10^{8}$ .

Write 0.0018 in standard form.

$0.0018 = 1.8 \times 10^(-3)$ .

Write $6.09 \times 10^{8}$ as an ordinary number.

$6.09 \times 10^{8} = 609000000$ .

Write $2.06 \times 10^(-2)$ as an ordinary number.

$2.06 \times 10^(-2) = 0.0206$ .

Calculate $(4.1 \times 10^(-3)) \times (8.9 \times 10^{7})$ and give your answer in standard form.

$4.1 \times 8.9 = 36.49$ , and $10^(-3) \times 10^{7} = 10^{4}$ , so the product is $36.49 \times 10^{4} = 3.649 \times 10^{5}$ .

Calculate $(1.63 \times 10^{12}) \times (2.47 \times 10^(-1))$ and give your answer in standard form.

$1.63 \times 2.47 = 4.0261$ , and $10^{12} \times 10^(-1) = 10^{11}$ , so the product is $4.0261 \times 10^{11}$ .

Calculate $(8.1 \times 10^{3}) + (7.9 \times 10^{4})$ and give your answer in standard form.

Convert to same power: $8.1 \times 10^{3} = 0.81 \times 10^{4}$ , then add: $(0.81 + 7.9) \times 10^{4} = 8.71 \times 10^{4}$ .

Calculate $(6.3 \times 10^{2}) \div (7 \times 10^(-3))$ and give your answer in standard form.

$6.3 \div 7 = 0.9$ , and $10^{2} \div 10^(-3) = 10^{5}$ , so the result is $0.9 \times 10^{5} = 9 \times 10^{4}$ .

Find the value of x when $5^{x} \times 5^{3} = 5^{12}$ .

Using the law $a^{m} \times a^{n} = a^(m+n)$ , we have $5^(x+3) = 5^{12}$ , so $x+3 = 12$ , hence $x = 9$ .

Find the value of p when $5^{p} \div 5^{8} = 5^{13}$ .

Using the law $a^{m} \div a^{n} = a^(m-n)$ , we have $5^(p-8) = 5^{13}$ , so $p-8 = 13$ , hence $p = 21$ .

Find the value of $2^(-4)$ without a calculator.

$2^(-4) = 1/(2^{4}) = \frac{1}{16}$ .

Find the value of $5^(-2)$ .

$5^(-2) = 1/(5^{2}) = \frac{1}{25}$ .

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