Algebraic Roots & Indices Flashcards

All 8 cards in this deck

What are the index laws for multiplying and for dividing powers of the same base?

e.g. $2x^3y^5 \times 7x^2y$ and $\dfrac{32q^9r^4}{4q^3r}$

Multiplying: add the indices; dividing: subtract the indices (coefficients are multiplied/divided as normal).

$14x^5y^6$ and $8q^6r^3$

True or false? $3^{-2} = -9$

False. A negative index means the reciprocal: $a^{-n} = \dfrac{1}{a^n}$, so $3^{-2} = \dfrac{1}{9}$.

What does $a^{\frac{m}{n}}$ mean?

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

The $n$th root of $a$, raised to the power $m$: $a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$.

$8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 4$

What are the steps to find an unknown index in an index equation?

e.g. $3^{x+1} = \dfrac{9^2}{27}$

1. Write every term as a power of a common base: $\dfrac{(3^2)^2}{3^3}$
2. Use the index laws to get a single power on each side: $3^{x+1} = 3^{4-3} = 3^1$
3. Equate the indices and solve: $x+1 = 1$, so $x = 0$

What is the index law for raising a power to a power?

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

$(a^m)^n = a^{mn}$ — multiply the indices.

$(p^2)^5 = p^{10}$

What are the steps to simplify a bracket with a coefficient raised to a power?

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

1. Raise the number to the power: $5^3 = 125$
2. Multiply each letter's index by the outside power: $n^3$ and $p^9$
3. Write as one product: $125n^3p^9$

What are the steps to simplify an algebraic square root?

e.g. $\sqrt{16x^6}$

1. Square root the number: $\sqrt{16} = 4$
2. Halve the index of each letter: $x^6 \to x^3$
3. Write as one product: $4x^3$

True or false? $\sqrt{9x^{16}} = 3x^8$

True. You square root the coefficient and halve the index.