IB · Unit 1 · 1A Exponents
A negative exponent means the reciprocal. Flip first, then make every exponent positive.
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Powers of 2
Each time the exponent drops by 1, divide by 2.
| Power | Value |
|---|---|
| \(2^{-4}\) | \(\dfrac{1}{16}\) |
| \(2^{-3}\) | \(\dfrac{1}{8}\) |
| \(2^{-2}\) | \(\dfrac{1}{4}\) |
| \(2^{-1}\) | \(\dfrac{1}{2}\) |
| \(2^{0}\) | \(1\) |
| \(2^{1}\) | \(2\) |
| \(2^{2}\) | \(4\) |
| \(2^{3}\) | \(8\) |
| \(2^{4}\) | \(16\) |
\(2^{3} = 8\) and \(2^{-3} = \dfrac{1}{8}\).
\(2^{-3}\) is the same as \(2^{3}\), written as a fraction: \(2^{-3} = \dfrac{1}{2^{3}} = \dfrac{1}{8}\).
Negative means reciprocal
Move the power across the fraction bar and make the exponent positive.
\(a^{-n} = \dfrac{1}{a^n}\)
Example:
\(3^{-2} = \dfrac{1}{3^{2}}\)
\(\dfrac{1}{9}\)
A fraction to a negative power
A negative exponent flips the fraction. Then apply the positive power.
\(\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}\)
Example:
\(\left(\dfrac{2}{5}\right)^{-1} = \dfrac{5}{2}\)
A number times a negative power
The coefficient stays in the numerator. Only the negative-power factor moves.
\(5n^{-1} = \dfrac{5}{n}\)
Power of a product
The outer exponent hits every factor, including negative exponents already inside.
\((ab)^{n} = a^{n} b^{n}\)
Example:
\((2x)^{3} = 2^{3} x^{3}\)
\(8x^{3}\)
A big fraction
Shortcut: a negative exponent flips to the other side of the fraction bar and becomes positive. Then simplify.
\(\dfrac{x^{-4} y^{2}}{x y^{-3}} = \dfrac{y^{2} y^{3}}{x x^{4}} = \dfrac{y^{5}}{x^{5}}\)
Write as a fraction in simplest terms.
A negative exponent means: write 1 over the same base with a positive exponent.
\(5^{-2} = \dfrac{1}{5^{2}}\)
\(\dfrac{1}{25}\)
Answer: \(\dfrac{1}{25}\)
A fraction to the −1 power means flip the fraction.
\(\left(\dfrac{2}{9}\right)^{-1} = \dfrac{9}{2}\)
Answer: \(\dfrac{9}{2}\)
\(8^{-1} = \dfrac{1}{8}\)
Answer: \(\dfrac{1}{8}\)
\(2^{-4} = \dfrac{1}{2^4} = \dfrac{1}{16}\)
Answer: \(\dfrac{1}{16}\)
\(x^{-3} = \dfrac{1}{x^3}\)
Answer: \(\dfrac{1}{x^3}\)
Flip the fraction.
\(\dfrac{4}{3}\)
Answer: \(\dfrac{4}{3}\)
Flip the fraction.
\(\dfrac{2}{5}\)
Answer: \(\dfrac{2}{5}\)
Keep the coefficient. Only the negative-power piece moves.
\(5^{-2} = \dfrac{1}{25}\)
\(4 \cdot \dfrac{1}{25} = \dfrac{4}{25}\)
Answer: \(\dfrac{4}{25}\)
A negative exponent flips the fraction.
\(\left(\dfrac{2}{3}\right)^{-2} = \left(\dfrac{3}{2}\right)^{2}\)
Then apply the power.
\(\left(\dfrac{3}{2}\right)^{2} = \dfrac{9}{4}\)
Answer: \(\dfrac{9}{4}\)
\(2^{-3} = \dfrac{1}{8}\)
\(9 \cdot \dfrac{1}{8} = \dfrac{9}{8}\)
Answer: \(\dfrac{9}{8}\)
The 8 stays on top. The k moves.
\(\dfrac{8}{k^2}\)
Answer: \(\dfrac{8}{k^2}\)
A negative exponent flips the fraction.
\(\left(\dfrac{3}{7}\right)^{-2} = \left(\dfrac{7}{3}\right)^{2}\)
Then apply the power.
\(\left(\dfrac{7}{3}\right)^{2} = \dfrac{49}{9}\)
Answer: \(\dfrac{49}{9}\)
\(\dfrac{12}{m}\)
Answer: \(\dfrac{12}{m}\)
The −6 is a number. Negative numbers are fine. Keep the negative with the 6.
Only the negative exponent needs to be rewritten.
\(\dfrac{-6}{x^{3}}\)
Answer: \(\dfrac{-6}{x^{3}}\)
The −10 stays with the 10 on top.
Only \(y^{-4}\) moves.
\(\dfrac{-10}{y^{4}}\)
Answer: \(\dfrac{-10}{y^{4}}\)
Distribute the outer exponent first. Then rewrite with positive exponents.
Square both factors.
\(4^2 \cdot (x^{-3})^2 = 16x^{-6}\)
\(16x^{-6} = \dfrac{16}{x^6}\)
Answer: \(\dfrac{16}{x^6}\)
\(2^3 \cdot (y^{-4})^3 = 8y^{-12}\)
\(\dfrac{8}{y^{12}}\)
Answer: \(\dfrac{8}{y^{12}}\)
\(5^{-1} \cdot (a^{-2})^{-1} = 5^{-1} a^{2}\)
\(\dfrac{a^2}{5}\)
Answer: \(\dfrac{a^2}{5}\)
A negative exponent flips the fraction, then apply the power.
\(\left(\dfrac{5q^3}{2p}\right)^{2} = \dfrac{25q^6}{4p^2}\)
Answer: \(\dfrac{25q^6}{4p^2}\)
A negative exponent flips the fraction, then apply the power.
\(\left(\dfrac{n}{3m^2}\right)^{3}\)
\(\dfrac{n^3}{27m^6}\)
Answer: \(\dfrac{n^3}{27m^6}\)
The outer exponent hits every factor.
\(6^{-2} \cdot (b^{-1})^{-2} \cdot (c^{2})^{-2}\)
Power to a power: multiply the exponents.
\(6^{-2} \cdot b^{2} \cdot c^{-4}\)
\(\dfrac{b^{2}}{6^{2} c^{4}}\)
\(\dfrac{b^{2}}{36 c^{4}}\)
Answer: \(\dfrac{b^{2}}{36 c^{4}}\)
Shortcut: a negative exponent flips to the other side of the fraction bar and becomes positive. First move every negative exponent. Then simplify.
Move every negative exponent to the other side. It becomes positive.
\(\dfrac{2^{3} \bbox[4px,#fde68a]{x^{-4}} y^{2}}{4 x \bbox[4px,#fde68a]{y^{-3}}}\)
\(\dfrac{2^{3} y^{2} y^{3}}{4 x x^{4}}\)
\(\dfrac{8 y^{5}}{4 x^{5}}\)
\(\dfrac{2 y^{5}}{x^{5}}\)
Answer: \(\dfrac{2 y^{5}}{x^{5}}\)
Move every negative exponent to the other side. It becomes positive.
\(\dfrac{-16 \bbox[4px,#fde68a]{s^{-2}} t^{3}}{4 s^{3} \bbox[4px,#fde68a]{t^{-1}}}\)
\(\dfrac{-16 t^{3} t}{4 s^{3} s^{2}}\)
\(\dfrac{-16 t^{4}}{4 s^{5}}\)
\(\dfrac{-4 t^{4}}{s^{5}}\)
Answer: \(\dfrac{-4 t^{4}}{s^{5}}\)
Move every negative exponent to the other side. It becomes positive.
\(\dfrac{2^{4} \bbox[4px,#fde68a]{a^{-5}} b^{2} \bbox[4px,#fde68a]{c^{-1}}}{8 \bbox[4px,#fde68a]{a^{-2}} b^{6} \bbox[4px,#fde68a]{c^{-1}}}\)
\(\dfrac{2^{4} a^{2} b^{2} c}{8 a^{5} b^{6} c}\)
\(\dfrac{16 a^{2} b^{2}}{8 a^{5} b^{6}}\)
\(\dfrac{2}{a^{3} b^{4}}\)
Answer: \(\dfrac{2}{a^{3} b^{4}}\)
Move every negative exponent to the other side. It becomes positive.
\(\dfrac{5^{2} \bbox[4px,#fde68a]{p^{-3}} \bbox[4px,#fde68a]{q^{-2}} r^{4}}{25 p^{2} \bbox[4px,#fde68a]{q^{-5}} \bbox[4px,#fde68a]{r^{-1}}}\)
\(\dfrac{5^{2} q^{5} r^{4} r}{25 p^{2} p^{3} q^{2}}\)
\(\dfrac{25 q^{5} r^{5}}{25 p^{5} q^{2}}\)
\(\dfrac{q^{3} r^{5}}{p^{5}}\)
Answer: \(\dfrac{q^{3} r^{5}}{p^{5}}\)
Distribute each outer exponent.
\(\dfrac{(-2)^{-3} (p^{-3})^{-3} (q^{2})^{-3}}{4^{2} p^{2} (q^{-1})^{2}}\)
Power to a power: multiply the exponents.
\(\dfrac{(-2)^{-3} p^{9} q^{-6}}{16 p^{2} q^{-2}}\)
Move every negative exponent to the other side. It becomes positive.
\(\dfrac{\bbox[4px,#fde68a]{(-2)^{-3}} p^{9} \bbox[4px,#fde68a]{q^{-6}}}{16 p^{2} \bbox[4px,#fde68a]{q^{-2}}}\)
\(\dfrac{p^{9} q^{2}}{(-2)^{3} \cdot 16 p^{2} q^{6}}\)
\((-2)^{3} = -8\)
\(\dfrac{p^{9} q^{2}}{-8 \cdot 16 p^{2} q^{6}}\)
\(\dfrac{p^{7}}{-128 q^{4}}\)
\(\dfrac{-p^{7}}{128 q^{4}}\)
Answer: \(\dfrac{-p^{7}}{128 q^{4}}\)
Distribute each outer exponent.
\(\dfrac{\bbox[4px,#99f6e4]{(3 a^{-2} b^{4})^{3}}}{\bbox[4px,#e9d5ff]{(3 a)^{-2}} \bbox[4px,#bfdbfe]{(9 a b^{2})^{2}}}\)
\(\bbox[4px,#99f6e4]{(3 a^{-2} b^{4})^{3} = 3^{3} (a^{-2})^{3} (b^{4})^{3}}\)
\(\bbox[4px,#e9d5ff]{(3 a)^{-2} = 3^{-2} a^{-2}}\)
\(\bbox[4px,#bfdbfe]{(9 a b^{2})^{2} = 9^{2} a^{2} (b^{2})^{2}}\)
Power to a power: multiply the exponents.
\(\bbox[4px,#99f6e4]{3^{3} (a^{-2})^{3} (b^{4})^{3} = 27 a^{-6} b^{12}}\)
\(\bbox[4px,#e9d5ff]{3^{-2} a^{-2}}\)
\(\bbox[4px,#bfdbfe]{9^{2} a^{2} (b^{2})^{2} = 81 a^{2} b^{4}}\)
\(\dfrac{\bbox[4px,#99f6e4]{27 a^{-6} b^{12}}}{\bbox[4px,#e9d5ff]{3^{-2} a^{-2}} \cdot \bbox[4px,#bfdbfe]{81 a^{2} b^{4}}}\)
Move every negative exponent to the other side. It becomes positive.
\(\dfrac{27 \bbox[4px,#fde68a]{a^{-6}} b^{12}}{\bbox[4px,#fde68a]{3^{-2}} \bbox[4px,#fde68a]{a^{-2}} \cdot 81 a^{2} b^{4}}\)
\(\dfrac{27 \cdot 3^{2} a^{2} b^{12}}{81 a^{6} a^{2} b^{4}}\)
\(\dfrac{27 \cdot 9 a^{2} b^{12}}{81 a^{8} b^{4}}\)
\(\dfrac{243 a^{2} b^{12}}{81 a^{8} b^{4}}\)
\(\dfrac{3 b^{8}}{a^{6}}\)
Answer: \(\dfrac{3 b^{8}}{a^{6}}\)
Five problems that mix Easy through Hard 2. Write every answer with positive exponents.
The negative is on the exponent. Write 1 over the same base with a positive exponent.
\(7^{-2} = \dfrac{1}{7^{2}}\)
\(7^{2} = 7 \times 7 = 49\)
\(\dfrac{1}{49}\)
Answer: \(\dfrac{1}{49}\)
A negative exponent flips the fraction.
\(\dfrac{9}{4}\)
Answer: \(\dfrac{9}{4}\)
The −14 is a number. Keep the negative with the 14 on top.
Only \(z^{-2}\) moves.
\(\dfrac{-14}{z^{2}}\)
Answer: \(\dfrac{-14}{z^{2}}\)
The outer exponent hits every factor.
\(5^{-2} \cdot (x^{-1})^{-2} \cdot (y^{2})^{-2}\)
Power to a power: multiply the exponents.
\(5^{-2} \cdot x^{2} \cdot y^{-4}\)
\(\dfrac{x^{2}}{5^{2} y^{4}}\)
\(\dfrac{x^{2}}{25 y^{4}}\)
Answer: \(\dfrac{x^{2}}{25 y^{4}}\)
Move every negative exponent to the other side. It becomes positive.
\(\dfrac{3^{2} \bbox[4px,#fde68a]{m^{-5}} n^{3}}{9 m^{2} \bbox[4px,#fde68a]{n^{-1}}}\)
\(\dfrac{3^{2} n^{3} n}{9 m^{2} m^{5}}\)
\(\dfrac{9 n^{4}}{9 m^{7}}\)
\(\dfrac{n^{4}}{m^{7}}\)
Answer: \(\dfrac{n^{4}}{m^{7}}\)
The outer exponent hits every factor.
\(4^{-2} \cdot (d^{-1})^{-2} \cdot (e^{3})^{-2}\)
Power to a power: multiply the exponents.
\(4^{-2} \cdot d^{2} \cdot e^{-6}\)
\(\dfrac{d^{2}}{4^{2} e^{6}}\)
\(\dfrac{d^{2}}{16 e^{6}}\)
Answer: \(\dfrac{d^{2}}{16 e^{6}}\)
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