Powers, Roots & Standard Form Flashcards

All 24 cards in this deck

What does $5^4$ mean?

$5\times5\times5\times5 = 625$.

What are the powers of 2 from $2^1$ to $2^{10}$?

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.

True or false? $3^4 = 4^3$.

False. $3^4 = 81$ but $4^3 = 64$.

What does $\sqrt[n]{a}$ mean?

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

The number that gives $a$ when multiplied by itself $n$ times.
$\sqrt[3]{64}=4$, because $4\times4\times4=64$.

What are the steps to estimate the square root of a number that is not a square number?

e.g. estimate $\sqrt{78}$

1. Find the square numbers either side: $64 < 78 < 81$
2. So the root is between $8$ and $9$
3. $78$ is close to $81$, so $\sqrt{78}\approx 8.8$

What is $a^m \times a^n$?

e.g. $3^7 \times 3^{-2}$

$a^{m+n}$ — add the indices.
$3^7 \times 3^{-2} = 3^5$

What is $a^m \div a^n$?

e.g. $10^{360} \div 10^{150}$

$a^{m-n}$ — subtract the indices.
$10^{360} \div 10^{150} = 10^{210}$

What is $(a^m)^n$?

e.g. $(5^2)^3$

$a^{mn}$ — multiply the indices.
$(5^2)^3 = 5^6$

True or false? $7^0 = 0$.

False. Any non-zero number to the power $0$ equals $1$, so $7^0 = 1$.

What does $a^{-n}$ mean?

e.g. $2^{-4}$

$\frac{1}{a^{n}}$ — the reciprocal of $a^n$.
$2^{-4} = \frac{1}{16}$

What are the steps to work out $a^{\frac{m}{n}}$?

e.g. $8^{\frac{2}{3}}$

1. The denominator is the root: $\sqrt[3]{8} = 2$
2. The numerator is the power: $2^2 = 4$
3. So $8^{\frac{2}{3}} = 4$ (if the index is negative, take the reciprocal at the end)

What form must a number be written in to be in standard form?

$A \times 10^{n}$, 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. $468\,000$

1. Put the decimal point after the first non-zero digit: $4.68$
2. Count the places the point has moved: $5$
3. Write it as $4.68 \times 10^{5}$

True or false? A number written in standard form with a negative power of 10 is a negative number.

False. A negative power means the number is between 0 and 1, e.g. $7 \times 10^{-3} = 0.007$.

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

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

1. Multiply the front numbers: $3 \times 4 = 12$
2. Add the powers of 10: $10^{5+(-2)} = 10^{3}$
3. Rewrite in standard form: $12 \times 10^{3} = 1.2 \times 10^{4}$

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 front numbers: $8 \div 2 = 4$
2. Subtract the powers of 10: $10^{7-3} = 10^{4}$
3. Check $A$ is between 1 and 10: $4 \times 10^{4}$

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

e.g. $9.7 \times 10^{6} + 2.45 \times 10^{7}$

1. Write both with the same power of 10: $9.7 \times 10^{6} + 24.5 \times 10^{6}$
2. Add the front numbers: $34.2 \times 10^{6}$
3. Write in standard form: $3.42 \times 10^{7}$

True or false? $(3 \times 10^{5}) + (4 \times 10^{5}) = 7 \times 10^{10}$.

False. When adding like this the power of 10 stays the same: $7 \times 10^{5}$.

How do you work out how many times bigger one number in standard form is than another?

e.g. $4.35 \times 10^{9}$ compared with $4.14 \times 10^{7}$

Divide the larger by the smaller: $(4.35 \times 10^{9}) \div (4.14 \times 10^{7}) = 105$ (to the nearest whole number), so it is over 100 times bigger.

True or false? Every positive number has two square roots, one positive and one negative.

True. e.g. $10^2 = 100$ and $(-10)^2 = 100$, so the square roots of 100 are $10$ and $-10$.

What are the steps to solve an equation of the form $a^x = b$ by equating indices?
e.g. solve $2^x = 32$

1. Write both sides as powers of the same base: $2^x = 2^5$
2. Since the bases are the same, equate the indices: $x = 5$

True or false? To solve $4^x = 8$ you can write both sides as powers of 2.

True. $4^x = 2^{2x}$ and $8 = 2^3$, so $2x = 3$.

What are the steps to find the average mass of one item when the total mass and the number of items are given in standard form?
e.g. $2 \times 10^{12}$ cells have a total mass of $90$ g

1. Divide the total mass by the number of items: $90 \div (2 \times 10^{12}) = 45 \times 10^{-12}$
2. Rewrite in standard form: $4.5 \times 10^{-11}$ g

True or false? A calculator display of $4.52\text{E}7$ means $4.52 \times 10^{7}$.

True. The E (or $\times 10^{x}$ output) gives the power of 10, so it should be written as $4.52 \times 10^{7}$.