Laws of Exponents and Algebraic Manipulation

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Laws of Exponents and Algebraic Manipulation — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionNegative and Zero ExponentsConcept Practice

2 marks~3 minCriterion A

A nanoscale sensor chip contains a square circuit element with side length

3^{-2}

Calculate the area of the circuit element, expressing your answer as a simplified fraction. [1]

A second square element has an area of

\dfrac{1}{6561}

^2

. Deduce whether its side length is greater than, equal to, or less than that of the first element, justifying your answer using index notation. [1]

Solutions

2QuestionNegative and Zero ExponentsAssessment Practice

2 marks~3 minCriterion C

A car currently has a value of 8000 dollars. Its value depreciates at a constant rate of 10% per year, modelled by

V = P(1 - r)^{t}

, where

V

is the value after

t

years,

P

is the initial value, and

r

is the annual depreciation rate.

Calculate the value of the car 2 years ago. [1]

A buyer argues that this model overestimates how much the car was worth 2 years ago. Critique this claim, with reference to both the mathematical result and the real-world depreciation pattern. [1]

Solutions

3QuestionNegative and Zero ExponentsAssessment Practice

4 marks~6 minCriterion B

A population of bacteria doubles every hour. A scientist models the population using

y = 2^x

, where

x

is the number of hours elapsed (negative values represent time before observation began) and

y

is the population in millions.

Selected values from the model:

x

-2

-1

0

1

2

y

(millions):

\dfrac{1}{4}

\dfrac{1}{2}

?

2

4

Explain how the pattern in the table demonstrates that

2^0 = 1

. [2]

A second bacterial strain follows

y = 5^x

and a third follows

y = (-3)^x

. Evaluate

5^0

and

(-3)^0

, justifying your answers using the zero exponent rule. [1]

The scientist claims the population at

x = 0

is "effectively zero" because no growth has yet occurred. Assess whether the mathematical model supports or contradicts this claim, and interpret what

2^0 = 1

means in this context. [1]

Solutions

4QuestionNegative and Zero ExponentsAssessment Practice

6 marks~9 minCriterion D

A patient receives a 500 mg dose of a drug. The amount of drug remaining in the body after

t

hours is modelled by

A = 500 \times 2^{-t},

where

A

is measured in mg. A second dose may be given once

A

falls below 100 mg.

Show that the amount of drug remaining after 3 hours is 62.5 mg. [2]

Deduce, using your result from part (a), whether a second dose can be given at

t = 3

hours. Justify your answer in the context of the model. [1]

Advise a clinician whether the model

A = 500 \times 2^{-t}

alone is sufficient to determine the safe timing of a second dose. In your response, discuss at least two real-world limitations and explain how neglecting these limitations could affect patient safety. [3]

Solutions

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