IB Mathematics Applications & Interpretation Topic…

Simplify the following expressions, giving your answers in the form $a x^{n}$ where $a$ and $n$ are rational numbers and any fractions are in lowest terms.

(a) $\left(\frac{2}{9} x^{\frac{1}{2}} \times \frac{1}{18} x^{\frac{-3}{4}}\right)^{-\frac{1}{4}}$

(b) $\frac{\left(8 x^{-\frac{2}{3}}\right)^{\frac{2}{3}}}{\left(64 x^{-\frac{1}{3}}\right)^{\frac{1}{3}}}$

A position sensor records a positive response $E_h(x)$ at numerical position $x$ metres. A source is at the unknown position $h$ metres. The sensor model is

$E_h(x)=\left(81(x-h)^{-4}\right)^{\frac34},\qquad x\ne h.$

The reference readings are $E_h(1)=1$ and $E_h(0)=\frac{27}{8}$ .

(c) Find all real solutions of $E_h(x)=1$ . A student instead simplifies the denominator to $(x+2)^3$ ; exactly why this loses a solution.

Two photographs have positive mean brightness values $B_1$ and $B_2$ . Their brightness ratio is $x=\frac{B_1}{B_2}\gt0$ . An image-comparison test gives

$x^{\frac12}+x^{-\frac12}=3.$

The comparison is unchanged when the photographs are exchanged.

A laboratory models the heat-loss rate $H$ watts of animals by

$H=cM^pt^q,$

where $M\gt0$ is the numerical mass in kilograms, $t\gt0$ is the numerical cooling time in hours, and $c\gt0$ . The constants $p$ and $q$ are unknown. The heat-loss rate per kilogram is $R=\frac HM$ .

Two exact experimental observations are made:

Multiplying $M$ by 8 and $t$ by $\frac14$ leaves $R$ unchanged.
Multiplying by 27 and by 9 multiplies by .

(b) For $s\gt 0$ , suppose $M$ is multiplied by $s^m$ and $t$ by $s^n$ . the condition on real $m$ and $n$ that leaves $R$ unchanged for every $s$ .

A material-testing model assigns a response index $R_p$ to a film specimen. The positive quantities $a$ and $b$ are its area and thickness, each expressed as a ratio to a reference value. The model is

$R_p(a,b)=\left(\dfrac{16a^{-\frac32}b^{\frac53}}{a^{\frac12}b^{-\frac13}}\right)^{-\frac34}\left(\dfrac ab\right)^p.$

where $p$ is a constant. Uniformly scaling the specimen by a length factor $\lambda\gt0$ replaces $(a,b)$ by $(\lambda^2a,\lambda b)$ . Tests show that this multiplies the response index by for every .

(b) For this value of $p$ , two specimens have the same value of $b$ and satisfy $R_2=8R_1$ . Find $\dfrac{a_2}{a_1}$ .

(a) $h$ , taking full account of the even root hidden in the exponent.

(b) For this value of $h$ , all $c\gt 0$ for which $E_h(x)=c$ has exactly one solution in the restricted domain $x\gt 0$ .

(a) Without finding $x$ , the exact value of $x^{\frac32}+x^{-\frac32}$ .

(b) the exact value of $\left|x^{\frac12}-x^{-\frac12}\right|$ and why the given information is insufficient to decide whether $x\gt 1$ .

(c) Animal B has $64$ times the mass of animal A. the factor by which its exposure time must change to keep $R$ unchanged, and why a proposed factor of $16$ has the wrong direction.

(a) the value of $p$ .

IB Mathematics Applications & Interpretation AHL 1.10—expressions with Non-integer Exponents Questionbank

IB Mathematics Applications & Interpretation AHL 1.10—expressions with Non-integer Exponents Questionbank

IB Mathematics Applications & Interpretation AHL 1.10—expressions with Non-integer Exponents Questionbank

IB Mathematics Applications & Interpretation AHL 1.10—expressions with Non-integer Exponents Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 1.10—expressions with Non-integer Exponents Questionbank