Sec 3 · G1 (N(T))

Indices, compound interest and standard form

Practise indices, compound interest and standard form with step-by-step help. Free account.

1. Zero, negative and fractional indices

Zero index

\displaystyle a^{0} = 1, \quad a \neq 0

e.g.

\displaystyle 7^{0} = 1

Negative index

\displaystyle a^{-n} = \dfrac{1}{a^{n}}

e.g.

\displaystyle 2^{-3}

\displaystyle = \dfrac{1}{2^{3}}

\displaystyle = \dfrac{1}{8}

Negative index of a fraction: flip it

\displaystyle \left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}

e.g.

\displaystyle \left(\dfrac{2}{3}\right)^{-2}

\displaystyle = \left(\dfrac{3}{2}\right)^{2}

\displaystyle = \dfrac{9}{4}

📖 Worked example

Evaluate $3^{-3}$ .

Step 1 of 3

Rewrite with a positive index

3^{-3} = \left(\dfrac{1}{3}\right)^{3}

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2. Compound interest

Compound interest

\displaystyle A = P\left(1 + \dfrac{r}{100}\right)^{n}

Interest earned

\displaystyle \text{Interest} = A - P

Depreciation (losing value)

\displaystyle A = P\left(1 - \dfrac{r}{100}\right)^{n}

Symbol	Meaning	Example
$A$	total amount at the end	$5624.32
$P$	amount invested (principal)	$5000
$r$	interest rate per annum, %	$4\%$ p.a.
$n$	number of years	$3$ years

Compounded	Rate per period	Number of periods
yearly	$r$	$n$
half-yearly	$\dfrac{r}{2}$	$2n$
quarterly	$\dfrac{r}{4}$	$4n$
monthly	$\dfrac{r}{12}$	$12n$

📖 Worked example

Mrs Tan invests $5000 at 3% per annum compound interest, compounded half-yearly. Find the total amount after 3 years, correct to the nearest cent.

Step 1 of 3

Use the compound interest formula, with the rate and the number of periods for each compounding period

A = P\left(1 + \dfrac{r}{100}\right)^n

A = 5000\left(1 + \dfrac{3}{100 \times 2}\right)^{6}

📖 Worked example

The value of a car bought for $25 000 depreciates at 15% per annum. Find its value after 2 years, correct to the nearest cent.

Step 1 of 2

Use the compound interest formula

A = P\left(1 - \dfrac{r}{100}\right)^n

A = 25000\left(1 - \dfrac{15}{100}\right)^{2}

Practise this →

3. Standard form

Standard form

\displaystyle A \times 10^{n}, \quad 1 \leq A < 10

$n$ is an integer

Number	Standard form	Point moves
$3\,450\,000$	$3.45 \times 10^{6}$	6 places left
$0.000\,345$	$3.45 \times 10^{-4}$	4 places right
$45\,000$	$4.5 \times 10^{4}$	4 places left
$0.07$	$7 \times 10^{-2}$	2 places right

Multiply: multiply the numbers, add the powers

\displaystyle (a \times 10^{m}) \times (b \times 10^{n}) = ab \times 10^{m + n}

e.g.

\displaystyle (4 \times 10^{3}) \times (5 \times 10^{2})

\displaystyle = 20 \times 10^{5}

\displaystyle = 2 \times 10^{6}

Divide: divide the numbers, subtract the powers

\displaystyle \dfrac{a \times 10^{m}}{b \times 10^{n}} = \dfrac{a}{b} \times 10^{m - n}

e.g.

\displaystyle (9 \times 10^{5}) \div (3 \times 10^{2})

\displaystyle = 3 \times 10^{3}

Add or subtract: same power of 10 first

\displaystyle a \times 10^{n} + b \times 10^{n} = (a + b) \times 10^{n}

e.g.

\displaystyle 3.2 \times 10^{4} + 6.4 \times 10^{3}

\displaystyle = 3.2 \times 10^{4} + 0.64 \times 10^{4}

\displaystyle = 3.84 \times 10^{4}

📖 Worked example

Express $0.000\,345$ in standard form.

Step 1 of 2

Move the decimal point to just after the first non-zero digit

Number of places the point moves:

4

📖 Worked example

Calculate $(3.2 \times 10^{4}) + (6.4 \times 10^{3})$ , giving your answer in standard form.

Step 1 of 2

Write both with the same power of 10

6.4 \times 10^{3} = 0.64 \times 10^{4}

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Sec 3 · G1 (N(T))

Indices, compound interest and standard form

Practise indices, compound interest and standard form with step-by-step help. Free account.

Start free

1. Zero, negative and fractional indices

Zero index

\displaystyle a^{0} = 1, \quad a \neq 0

e.g.

\displaystyle 7^{0} = 1

Negative index

\displaystyle a^{-n} = \dfrac{1}{a^{n}}

e.g.

\displaystyle 2^{-3}

\displaystyle = \dfrac{1}{2^{3}}

\displaystyle = \dfrac{1}{8}

Negative index of a fraction: flip it

\displaystyle \left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}

e.g.

\displaystyle \left(\dfrac{2}{3}\right)^{-2}

\displaystyle = \left(\dfrac{3}{2}\right)^{2}

\displaystyle = \dfrac{9}{4}

📖 Worked example

Evaluate $3^{-3}$ .

Step 1 of 3

Rewrite with a positive index

3^{-3} = \left(\dfrac{1}{3}\right)^{3}

Practise this →

2. Compound interest

Compound interest

\displaystyle A = P\left(1 + \dfrac{r}{100}\right)^{n}

Interest earned

\displaystyle \text{Interest} = A - P

Depreciation (losing value)

\displaystyle A = P\left(1 - \dfrac{r}{100}\right)^{n}

Symbol	Meaning	Example
$A$	total amount at the end	$5624.32
$P$	amount invested (principal)	$5000
$r$	interest rate per annum, %	$4\%$ p.a.
$n$	number of years	$3$ years

Compounded	Rate per period	Number of periods
yearly	$r$	$n$
half-yearly	$\dfrac{r}{2}$	$2n$
quarterly	$\dfrac{r}{4}$	$4n$
monthly	$\dfrac{r}{12}$	$12n$

📖 Worked example

Mrs Tan invests $5000 at 3% per annum compound interest, compounded half-yearly. Find the total amount after 3 years, correct to the nearest cent.

Step 1 of 3

Use the compound interest formula, with the rate and the number of periods for each compounding period

A = P\left(1 + \dfrac{r}{100}\right)^n

A = 5000\left(1 + \dfrac{3}{100 \times 2}\right)^{6}

📖 Worked example

The value of a car bought for $25 000 depreciates at 15% per annum. Find its value after 2 years, correct to the nearest cent.

Step 1 of 2

Use the compound interest formula

A = P\left(1 - \dfrac{r}{100}\right)^n

A = 25000\left(1 - \dfrac{15}{100}\right)^{2}

Practise this →

3. Standard form

Standard form

\displaystyle A \times 10^{n}, \quad 1 \leq A < 10

$n$ is an integer

Number	Standard form	Point moves
$3\,450\,000$	$3.45 \times 10^{6}$	6 places left
$0.000\,345$	$3.45 \times 10^{-4}$	4 places right
$45\,000$	$4.5 \times 10^{4}$	4 places left
$0.07$	$7 \times 10^{-2}$	2 places right

Multiply: multiply the numbers, add the powers

\displaystyle (a \times 10^{m}) \times (b \times 10^{n}) = ab \times 10^{m + n}

e.g.

\displaystyle (4 \times 10^{3}) \times (5 \times 10^{2})

\displaystyle = 20 \times 10^{5}

\displaystyle = 2 \times 10^{6}

Divide: divide the numbers, subtract the powers

\displaystyle \dfrac{a \times 10^{m}}{b \times 10^{n}} = \dfrac{a}{b} \times 10^{m - n}

e.g.

\displaystyle (9 \times 10^{5}) \div (3 \times 10^{2})

\displaystyle = 3 \times 10^{3}

Add or subtract: same power of 10 first

\displaystyle a \times 10^{n} + b \times 10^{n} = (a + b) \times 10^{n}

e.g.

\displaystyle 3.2 \times 10^{4} + 6.4 \times 10^{3}

\displaystyle = 3.2 \times 10^{4} + 0.64 \times 10^{4}

\displaystyle = 3.84 \times 10^{4}

📖 Worked example

Express $0.000\,345$ in standard form.

Step 1 of 2

Move the decimal point to just after the first non-zero digit

Number of places the point moves:

4

📖 Worked example

Calculate $(3.2 \times 10^{4}) + (6.4 \times 10^{3})$ , giving your answer in standard form.

Step 1 of 2

Write both with the same power of 10

6.4 \times 10^{3} = 0.64 \times 10^{4}

Practise this →

Practise indices, compound interest and standard form with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

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