Indices Flashcards

All 6 cards in this deck

True or false? $x^3 \times x^4 = x^{12}$

False. Multiplying powers of the same base means adding the indices: $x^3 \times x^4 = x^7$.

What is $3x^{-7}$ written as a fraction?

$\frac{3}{x^{7}}$

The negative index only makes the reciprocal — it does not make the term negative, and the $3$ stays on top.

What does the fractional index $x^{\frac{a}{b}}$ mean?

e.g. $y^{\frac{5}{6}}$

$\sqrt[b]{x^{a}}$ — the $b$th root of $x^{a}$.

e.g. $y^{\frac{5}{6}} = \sqrt[6]{y^{5}}$, and $\sqrt{y} = y^{\frac{1}{2}}$, $\sqrt[3]{y} = y^{\frac{1}{3}}$.

What are the steps to simplify an algebraic fraction with a bracketed power?

e.g. simplify fully $\frac{12w^{8}}{(4w^{3})^{2}}$

1. Apply the power to everything in the bracket: $(4w^{3})^{2} = 16w^{6}$
2. Simplify the numbers: $\frac{12}{16} = \frac{3}{4}$
3. Subtract the indices: $w^{8-6} = w^{2}$, giving $\frac{3w^{2}}{4}$

What are the steps to solve an index equation with different bases?

e.g. solve $2^{4x+1} = 8^{x-1}$

1. Write both sides as powers of the same base: $8^{x-1} = (2^{3})^{x-1} = 2^{3x-3}$
2. Equate the indices: $4x + 1 = 3x - 3$
3. Solve the resulting equation: $x = -4$

True or false? The equation $x^{\frac{2}{3}} = 9$ has only one solution.

False. It has two: $x = 27$ and $x = -27$, because the cube root of a negative number is negative and squaring removes the sign.