Laws of indices Flashcards

All 8 cards in this deck

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 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 $2x = 3$.