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**: 3⁰ $= 1$ .
Any number to the power of **1** equals itself: 3¹ $= 3$ .
**Square roots** reverse squaring; every positive number has two square roots (positive and negative). Notation √ refers to the positive root.
**Cube roots** reverse cubing; each number has exactly one real cube root (∛).
**nth roots**: if n is even, positive numbers have two real nth roots; negative numbers have none. If n is odd, every number has one real nth root.
**Reciprocal** of a number is 1 divided by that number; written as index **-1** (e.g., 5⁻¹ $= \frac{1}{5})$ .

Laws of Indices

**Multiplication**: aᵐ × aⁿ = aᵐ⁺ⁿ (add powers).
**Division**: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract powers).
**Power of a power**: (aᵐ)ⁿ = aᵐⁿ (multiply powers).
**Negative power**: a⁻ⁿ = 1/aⁿ (reciprocal).
**Fractional power**: $a^(\frac{1}{n}) =$ ⁿ√a; $a^(\frac{m}{n}) =$ (ⁿ√a)ᵐ = ⁿ√(aᵐ).
**Zero power**: a⁰ $= 1 (for a \ne 0)$ .
**Product and quotient rules**: (ab)ⁿ = aⁿbⁿ; (a/b)ⁿ = aⁿ/bⁿ.

Converting to Standard Form

Standard form: **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. Example: $32400 = 3.24 \times$ 10⁴.
For **small numbers** $(0 <$ number $< 1), n$ is **negative**: count how many places the decimal moves right. Example: $0.0000324 = 3.24 \times$ 10⁻⁵.
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

**Multiplication**: multiply the A parts, add the powers of 10, then adjust to standard form. Example: (3×10²)×(4×10⁵)=12×10⁷=1.2×10⁸.
**Division**: divide the A parts, subtract the powers of 10, then adjust. Example: (2×10⁻⁵)÷(8×10⁻³)=0.25×10⁻²=2.5×10⁻³.
**Addition/Subtraction (Method 1)**: convert both to ordinary numbers, add/subtract, then convert back to standard form.
**Addition/Subtraction (Method 2)**: rewrite numbers to have the same power of 10, add/subtract the A parts, then adjust. Efficient for large or negative exponents.
On a calculator, use brackets and the ×10ˣ button to enter standard form numbers.

Common Mistakes & Tips

Index laws only work with **same base**; $e.g., 2^{3}\times 5^{2}$ cannot be simplified directly.
A negative power means reciprocal, not a negative number: 2⁻⁴ $= \frac{1}{16}$ , not -16.
When taking even roots of positive numbers, remember there are **two** roots $(e.g., \sqrt{25} = \pm 5)$ , but the radical sign denotes the principal (positive) root.
In standard form, A must be **≥1 and <10**; if not, adjust by moving the decimal and changing the exponent accordingly.

Powers and Roots Relationship

Standard Form Conversion Steps

Multiplying in Standard Form

Index Laws Summary

Practice questions

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1.Write 510 100 000 in standard form.
Easy
- A $5.101 \times 10^{8}$
- B $5.101 \times 10^{9}$
- C $51.01 \times 10^{7}$
- D $5.101 \times 10^{7}$
2.Write 0.00527 in standard form.
Easy
- A $5.27 \times 10^{-3}$
- B $5.27 \times 10^{-4}$
- C $5.27 \times 10^{3}$
- D $52.7 \times 10^{-4}$
3.Write $3.4 \times 10^{-1}$ as an ordinary number.
Easy
- A0.34
- B34
- C3.4
- D0.034
4.Which of these numbers is the largest? $3.4 \times 10^{-1}, 1.36 \times 10^{6}, 7.9 \times 10^{0}, 2.4 \times 10^{5}, 5.21 \times 10^{-3}, 4.3 \times 10^{-2}$
Easy
- A $1.36 \times 10^{6}$
- B $2.4 \times 10^{5}$
- C $7.9 \times 10^{0}$
- D $3.4 \times 10^{-1}$
5.Find the value of $(\frac{2}{5})^{2}$ .
Easy
- A $\frac{4}{25}$
- B $\frac{2}{25}$
- C $\frac{4}{5}$
- D $\frac{2}{5}$
6.Work out $(5.2 \times 10^{7}) + (5.2 \times 10^{6})$ . Give your answer in standard form.
Medium
- A $5.72 \times 10^{7}$
- B $5.2 \times 10^{7}$
- C $5.72 \times 10^{6}$
- D $1.04 \times 10^{8}$
7.Work out $(9 \times 10^{-4}) \times (3 \times 10^{7})$ . Give your answer in standard form.
Medium
- A $2.7 \times 10^{4}$
- B $2.7 \times 10^{3}$
- C $2.7 \times 10^{5}$
- D $27 \times 10^{3}$
8.Patrick says that $64^{\frac{1}{4}} = 16$ because $\frac{1}{4}$ of 64 is 16. What is wrong with his reasoning?
Medium
- A $64^{\frac{1}{4}}$ means the fourth root of 64, not $\frac{1}{4}$ of 64
- B $64^{\frac{1}{4}} = 64 \div 4 = 16$
- C $64^{\frac{1}{4}} = 64 \times \frac{1}{4} = 16$
- DNothing, Patrick is correct

Summary

## Powers, Roots & Standard Form ### Powers & Roots - **Powers (indices)**: base raised to exponent. e.g., 6³ = 6×6×6. - **Any non-zero number to power 0 = 1**: 3⁰ = 1. - **Any number to power 1 = itself**: 3¹ = 3. - **Square roots**: √25 = 5 (positive root). Every positive number has two square roots (±). Negative numbers have no real square root. - **Cube roots**: ∛125 = 5. Every number has one real cube root. - **nth roots**: For even n, positive and negative roots exist; negative numbers have no real nth 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 - (a/b)⁻ⁿ = (b/a)ⁿ = bⁿ/aⁿ - a^(1/n) = ⁿ√a - a^(-1/n) = 1/(ⁿ√a) - a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ) ### Standard Form - **Form**: A × 10ⁿ where 1 ≤ A < 10 and n is integer. - **Large numbers**: Move decimal left, n positive. e.g., 32400 = 3.24 × 10⁴. - **Small numbers**: Move decimal right, n negative. e.g., 0.0000324 = 3.24 × 10⁻⁵. ### Operations with Standard Form - **Multiply**: Multiply A's, add exponents. e.g., (3×10²)×(4×10⁵)=12×10⁷=1.2×10⁸. - **Divide**: Divide A's, subtract exponents. e.g., (2×10⁻⁵)÷(8×10⁻³)=0.25×10⁻²=2.5×10⁻³. - **Add/Subtract**: Convert to same power or to ordinary numbers. e.g., (3.2×10³)+(2.1×10²)=3200+210=3410=3.41×10³.

Key terms

Standard form: A number written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
Index (plural indices): The small raised number indicating the power; also called exponent.
Base: The big number that is raised to a power.
Reciprocal: 1 divided by the number; a number and its reciprocal multiply to 1.
Square root: A value that when squared equals the original number; √ denotes the positive root.
Cube root: A value that when cubed equals the original number; ∛ denotes the real cube root.
nth root: A value that when raised to the power n equals the original number; ⁿ√ denotes the positive root for even n.

Slides

Powers & Roots

Powers (or indices) show repeated multiplication: $6^{3} = 6 \times 6 \times 6$ .
Any non-zero number to the power of 0 equals 1: 3⁰ $= 1$ .
Any number to the power of 1 equals itself: 3¹ $= 3$ .
Square roots reverse squaring: $\sqrt{25} = 5 (the$ positive root).
Cube roots reverse cubing: ∛125 $= 5$ .

What is the value of 5⁰?

0
1
5
undefined

Laws of Indices

aᵐ × aⁿ = aᵐ⁺ⁿ (add powers when multiplying same base).
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract powers when dividing same base).
(aᵐ)ⁿ = aᵐⁿ (multiply powers when raising a power).
a⁻ⁿ = 1/aⁿ (negative power means reciprocal).
$a^(\frac{1}{n}) =$ ⁿ√a (fractional power 1/n means nth root).

Simplify $4^{3} \times$ 4⁵ $= 4^____$

Negative and Fractional Powers

a⁻ⁿ = 1/aⁿ, e.g. 2⁻⁴ $= \frac{1}{16}$ .
$a^(\frac{m}{n}) =$ (ⁿ√a)ᵐ or ⁿ√(aᵐ), e.g. $8^(\frac{2}{3}) =$ (∛8)² $= 2^{2} = 4$ .
$a^(-\frac{m}{n}) = 1 / a^(\frac{m}{n})$ .

Find the value of $64^(\frac{2}{3})$ .

Converting to Standard Form

Standard form: $A \times$ 10ⁿ where $1 \le A < 10$ and n is an integer.
For large numbers, move decimal left: $324 500 000 = 3.245 \times$ 10⁸.
For small numbers, move decimal right: $0.007052 = 7.052 \times$ 10⁻³.

Write 0.00527 in standard form: $____ \times 10^____$

Converting from Standard Form

To convert from standard form to ordinary number, move the decimal point n places.
If n is positive, move right: $3.245 \times$ 10⁸ $= 324 500 000$ .
If n is negative, move left: $7.052 \times$ 10⁻³ $= 0.007052$ .

Write $4.82 \times$ 10⁻³ as an ordinary number.

0.00482
0.0482
4820
0.000482

Multiplying in Standard Form

Multiply the A parts and add the powers of 10.
Example: $(3 \times 10^{2}) \times (4 \times$ 10⁵) $= 12 \times$ 10⁷ $= 1.2 \times$ 10⁸.
If $A \ge 10$ , adjust: $12 = 1.2 \times$ 10¹, then add exponents.

Work out $(9 \times$ 10⁻⁴) $\times (3 \times$ 10⁷) $= ____ \times 10^____$

Dividing in Standard Form

Divide the A parts and subtract the powers of 10.
Example: $(2 \times$ 10⁻⁵) $\div (8 \times$ 10⁻³) $= 0.25 \times$ 10⁻² $= 2.5 \times$ 10⁻³.
Be careful with negative exponents: subtracting a negative adds.

Calculate $(3 \times$ 10⁷) $\div (6 \times$ 10⁴). Give answer in standard form.

$5 \times 10^{2}$
$5 \times 10^{3}$
$0.5 \times 10^{3}$
$5 \times$ 10⁴

Adding and Subtracting in Standard Form

Method 1: Convert to ordinary numbers, add/subtract, then convert back.
Method 2: Adjust powers to match, then add/subtract the A parts.
Example: $(4 \times$ 10⁵⁰) $- (2 \times$ 10⁴⁸) $= 400 \times$ 10⁴⁸ $- 2 \times$ 10⁴⁸ $= 398 \times$ 10⁴⁸ $= 3.98 \times$ 10⁵⁰.

Which of these are true?

$5.2 \times$ 10⁷ $+ 5.2 \times$ 10⁶ $= 5.72 \times$ 10⁷
$0.038 \times 10^{2} = 3.8$
$3800 \times$ 10⁻⁴ $= 0.38$
$0.38 \times$ 10⁻¹ $= 3.8$

Summary of Key Rules

Index laws work only with same base.
Standard form: $A \times$ 10ⁿ with $1 \le A < 10$ .
Multiplying: multiply A, add exponents.
Dividing: divide A, subtract exponents.
Adding/Subtracting: match exponents first.

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