Laws of Exponents and Algebraic Manipulation

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Laws of Exponents and Algebraic Manipulation — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionNegative and Zero ExponentsConcept Practice

2 marks~3 minCriterion A

A microchip component has a rectangular cross-section with width

2^{-3}

m and length

2^{-2}

m. An engineer must confirm the component meets a size specification requiring an area less than

2^{-4}

m².

Calculate the area of the cross-section. [1]

Justify whether the component meets the size specification. [1]

Solutions

2QuestionNegative 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

3QuestionProduct Quotient and Power LawsAssessment Practice

4 marks~6 minCriterion C

The functions

y = 2^x

and

y = 2^{x+3}

are defined for all real

x

State the transformation that maps the graph of

y = 2^x

onto the graph of

y = 2^{x+3}

. [1]

The point

(0, 1)

lies on

y = 2^x

. Show that the image of this point under the transformation confirms your answer to part (a). [1]

A student claims: "Adding 3 inside the exponent is equivalent to multiplying the original function by a constant, so the transformation could also be described as a vertical stretch." Justify whether this claim is correct, using algebraic reasoning. [2]

Solutions

4QuestionWord Problems Involving ExponentsAssessment Practice

4 marks~6 minCriterion D

A student invests 5000 dollars in a savings account at a 6% annual interest rate compounded monthly. The account balance after

t

years is modelled by

A = P\!\left(1 + \frac{r}{n}\right)^{nt}

where

P

is the principal,

r

is the annual interest rate,

n

is the number of compounding periods per year, and

A

is the balance.

Show that the predicted balance after 5 years is approximately 6744.25 dollars. [1]

The student's goal is to reach a balance of 7000 dollars. Deduce the number of complete years required for the balance to exceed 7000 dollars. [1]

Advise the student whether this exponential model alone is a sufficient basis for making an investment decision. Refer to at least two specific limitations in your response. [2]

Solutions

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