A bacterium that doubles every hour, a medicine that loses half its effect after a certain time, a sheet of paper folded twenty times: all of these are written using powers. Until now the exponent was a natural number that counted "how many times to multiply". Now the exponent can be $0$ or negative, and the meaning changes: instead of multiplying, it is going to divide.
Look at what happens if you divide powers with the same base by subtracting the exponents, even when the result is $0$:
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Calculate the numerical value of the following powers with negative exponents, expressing the result as an irreducible fraction or an integer: $3^{-2}$, $(-2)^{-3}$,…
Calculate the numerical value of the following powers with negative exponents, expressing the result as an irreducible fraction or an integer: $3^{-2}$, $(-2)^{-3}$, $5^{-1}$ and $(-1)^{-4}$.
Analyse and discuss with reasoning the resulting sign of the mathematical expression $(-a)^{-n}$ depending on two factors: whether the real number $a$ is strictly…
Analyse and discuss with reasoning the resulting sign of the mathematical expression $(-a)^{-n}$ depending on two factors: whether the real number $a$ is strictly positive or negative, and whether the integer $n$ is even or odd. Draw up a chart or table showing the four possible cases resulting from this combination, accompanied by a specific numerical example for each case.
You can do it on paper and send a photo with your answer.
When the base is already a fraction, the negative exponent swaps the numerator and the denominator at the same time.
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Express the following powers with fractional bases using a positive exponent and then calculate their exact value: $\left(\frac{2}{5}\right)^{-3}$ and…
Express the following powers with fractional bases using a positive exponent and then calculate their exact value: $\left(\frac{2}{5}\right)^{-3}$ and $\left(-\frac{1}{4}\right)^{-2}$.
Combining the laws of indices with negative exponents
The laws you already know (product, quotient, power of a power) work in exactly the same way when the exponents are negative: you just have to add and subtract with their signs, carefully.
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Simplify the following fraction by expressing both the numerator and denominator as powers of the same prime base, and give the result as a single power:…
Simplify the following fraction by expressing both the numerator and denominator as powers of the same prime base, and give the result as a single power: $\frac{8^4}{4^5}$.
Simplify the following expression, first factorising the composite bases and applying the laws of powers with integer exponents. Leave the result as a product of powers…
Simplify the following expression, first factorising the composite bases and applying the laws of powers with integer exponents. Leave the result as a product of powers of prime numbers:
In a mathematics exam, a student solved a calculation by writing $2^{-3} \cdot 5^{-3} = 10^{-6}$. Explain in detail the student's conceptual error, state the correct law…
In a mathematics exam, a student solved a calculation by writing $2^{-3} \cdot 5^{-3} = 10^{-6}$. Explain in detail the student's conceptual error, state the correct law that should have been used, and calculate the true result as a fraction.
Multiplying by a power with a negative exponent is "splitting into parts": each time one unit of time passes, the quantity is divided by the same factor.
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In a chemical laboratory, a toxic substance contained in a liquid is being filtered. Each filter installed in the circuit reduces the amount of toxins to one third. If…
In a chemical laboratory, a toxic substance contained in a liquid is being filtered. Each filter installed in the circuit reduces the amount of toxins to one third. If the initial amount of toxins is $81$ grams, express as a power with a negative exponent the amount of toxin remaining after passing through $5$ successive filters and calculate its exact value.
Find the integer value of $x$ that satisfies the following exponential equation, expressing all terms as powers of the same base: $5^{x+2} \cdot 25^{-1} = 125^3$
Find the integer value of $x$ that satisfies the following exponential equation, expressing all terms as powers of the same base:
Back to the beginning: If a medicine loses half of its effect every hour, after $4$ hours the original dose will have been multiplied by $2^{-4} = \frac{1}{2^4} = \frac{1}{16}$.