Powers, Roots & Standard Form Flashcards

All 24 cards in this deck

What does $5^3$ mean?

$5 \times 5 \times 5 = 125$

What does $\sqrt{n}$ mean?

e.g. $\sqrt{64}$

The number that multiplies by itself to give $n$.
$\sqrt{64} = 8$

What does $\sqrt[3]{n}$ (the cube root) mean?

e.g. $\sqrt[3]{64}$

The number that multiplied by itself three times gives $n$.
$\sqrt[3]{64} = 4$

List the powers of $2$ from $2^1$ to $2^6$.

$2, 4, 8, 16, 32, 64$

True or false? $3^2$ means $3 \times 2$.

False. $3^2 = 3 \times 3 = 9$; the index tells you how many times to multiply the number by itself.

What are the steps to work out an expression containing powers and roots?

e.g. $3 + 2^2 \times \sqrt{9}$

1. Work out powers and roots first: $2^2 = 4$, $\sqrt{9} = 3$
2. Then multiply or divide: $4 \times 3 = 12$
3. Then add or subtract: $3 + 12 = 15$

What is the multiplication law of indices?

e.g. $2^5 \times 2^4$

$a^m \times a^n = a^{m+n}$ (add the indices).
$2^5 \times 2^4 = 2^9$

What is the division law of indices?

e.g. $2^5 \div 2^3$

$a^m \div a^n = a^{m-n}$ (subtract the indices).
$2^5 \div 2^3 = 2^2$

What is the power law of indices?

e.g. $(2^3)^2$

$(a^m)^n = a^{mn}$ (multiply the indices).
$(2^3)^2 = 2^6$

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

e.g. $7^0$

$1$
$7^0 = 1$

What does a negative index mean?

e.g. $5^{-2}$

$a^{-n} = \dfrac{1}{a^n}$ (the reciprocal).
$5^{-2} = \dfrac{1}{25}$

What are the steps to find a missing index?

e.g. $8^3 \times 8^x = 8^9$

1. Same base, so add the indices on the left: $3 + x$
2. Set the indices equal: $3 + x = 9$
3. Solve: $x = 6$

What form must a number written in standard form take?

$A \times 10^n$, where $1 \le A < 10$ and $n$ is an integer.

What are the steps to write a large ordinary number in standard form?

e.g. $438\,000$

1. Put the decimal point after the first non-zero digit: $4.38$
2. Count how many places the point moved: $5$
3. Number is $\ge 10$, so the power is positive: $4.38 \times 10^{5}$

In standard form, is the power of $10$ positive or negative for a number less than $1$?

e.g. $0.007$

Negative.
$0.007 = 7 \times 10^{-3}$

What are the steps to write a number in standard form as an ordinary number?

e.g. $3.42 \times 10^{4}$

1. Look at the power: $4$, and it is positive
2. Move the decimal point that many places right (left if negative): $34200$
3. Fill any gaps with zeros: $34\,200$

True or false? $0.6 \times 10^{5}$ is written in standard form.

False. $A$ must satisfy $1 \le A < 10$; in standard form it is $6 \times 10^{4}$.

How do you order numbers written in standard form?

e.g. $5.6 \times 10^{4}$ and $9 \times 10^{3}$

Compare the powers of $10$ first; if they are equal, compare the $A$ parts.
$9 \times 10^{3} < 5.6 \times 10^{4}$

What are the steps to multiply two numbers in standard form?

e.g. $(3 \times 10^{5}) \times (2 \times 10^{4})$

1. Multiply the number parts: $3 \times 2 = 6$
2. Add the powers of $10$: $10^{5+4} = 10^{9}$
3. Check $1 \le A < 10$: $6 \times 10^{9}$

What are the steps to divide two numbers in standard form?

e.g. $(8 \times 10^{7}) \div (2 \times 10^{3})$

1. Divide the number parts: $8 \div 2 = 4$
2. Subtract the powers of $10$: $10^{7-3} = 10^{4}$
3. Check $1 \le A < 10$: $4 \times 10^{4}$

What must you do if your answer comes out as $18 \times 10^{-4}$?

Adjust it so $1 \le A < 10$: $1.8 \times 10^{-3}$.

What are the steps to add two numbers in standard form with different powers?

e.g. $3 \times 10^{5} + 4 \times 10^{4}$

1. Write both as ordinary numbers: $300\,000$ and $40\,000$
2. Add them: $340\,000$
3. Write back in standard form: $3.4 \times 10^{5}$

True or false? To work out $2 \times 10^{3} + 3 \times 10^{3}$ you add the number parts and add the powers.

False. The powers stay the same when adding: the answer is $5 \times 10^{3}$.

True or false? A distance of $4 \times 10^{8}$ m is $100$ times greater than a distance of $4 \times 10^{6}$ m.

True. The power of $10$ is $2$ greater, so it is $10^2 = 100$ times bigger.