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AHL 1.10—Expressions with non-integer exponents - IB Questionbank | RevisionDojo

Practice AHL 1.10—Expressions with non-integer exponents with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

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.

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

[2]

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

[3]

Question 2

HLPaper 1

Questions on algebra and surds.

Solve the equation $2-x \sqrt{3}=\frac{7 x}{\sqrt{3}}$ , giving your answer in the form $\frac{\sqrt{a}}{b}$ , where $a, b \in \mathbb{Z}$ . Find the value of $a$ and the value of $b$ .

[3]

Given that $8 \sqrt{2}=2^{k}$ , find the value of $k$ .

[2]

Question 3

HLPaper 1

Simplify the following expressions. Express the final answer in numerator form only (using negative exponents where necessary).

$\frac{\left(4 x y^{-2}\right)\left(-12 x^{-4} y^{12}\right)}{6 x^{2} y}$

[3]

$\left(2 x^{-1} y^{-2}\right)^{-3}\left(4 x^{2} y^{3}\right)^{4}$

[3]

Question 4

HLPaper 1

Simplify the following expressions.

$\sqrt{\left(9 x^{6} y^{-2} z^{4}\right)^{3}}(3 x y z)^{-2}$

[3]

$\left(\frac{x^{2} z^{4}}{2 y^{5}}\right)\left(\frac{3 x^{2} y^{3}}{z^{2}}\right)^{2}$

[3]

Question 5

HLPaper 2

Consider the expression $x^{0.75} \cdot y^{1.25}$ .

Write the expression $\log(x^{0.75} \cdot y^{1.25})$ in terms of $\log(x)$ and $\log(y)$ .

[2]

If $x = 16$ and $y = 81$ , evaluate $x^{0.75} \cdot y^{1.25}$ .

[2]

Question 6

HLPaper 2

The volume $V$ (in cubic metres) of a spherical balloon being inflated is given by $V = kt^{2.5}$ , where $t$ is the time in minutes and $k$ is a constant. When $t = 2$ , the volume is 16 cubic metres.

Find the value of $k$ .

[2]

Calculate the volume after 5 minutes.

[2]

Determine the rate of change of volume with respect to time when $t = 3$ .

[3]

Question 7

HLPaper 1

Consider the expression $a^{\frac{1}{5}} \times a^{\frac{2}{5}}$ , which can be written in the form $a^{p}$ for $a > 0$ .

Find the value of $p$ .

[2]

Given that $a^{\frac{1}{5}} \times a^{\frac{2}{5}} = 8$ , find the value of $a$ .

[2]

Question 8

HLPaper 2

A biologist is studying the growth of a particular bacteria population in a lab. The population of the bacteria, $P(t)$ , at time $t$ (in hours) is modeled by the function $P(t) = 100e^{0.3t}$ .

Find the population of the bacteria after 5 hours.

[2]

Determine the time it takes for the population to double.

[2]

Question 9

HLPaper 2

Express the following in the form $x^{r}$

$(\sqrt[5]{x})^{6}$

[2]

$\sqrt[4]{\sqrt[3]{x}}$

[2]

$\frac{1}{\sqrt[3]{x^{4}}}$

[2]

Question 10

HLPaper 2

A scientist is studying the growth of a bacterial culture. The population $P$ (in thousands) after $t$ hours can be modeled by the equation $P(t) = 2.5t^{1.4} + 3$ .

Find the population after 2 hours.

[2]

Calculate the time when the population reaches 20 thousand bacteria.

[3]

Paper

Level

Question 1

HLPaper 1

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.

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

[2]

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

[3]

Question 2

HLPaper 1

Questions on algebra and surds.

Solve the equation $2-x \sqrt{3}=\frac{7 x}{\sqrt{3}}$ , giving your answer in the form $\frac{\sqrt{a}}{b}$ , where $a, b \in \mathbb{Z}$ . Find the value of $a$ and the value of $b$ .

[3]

Given that $8 \sqrt{2}=2^{k}$ , find the value of $k$ .

[2]

Question 3

HLPaper 1

Simplify the following expressions. Express the final answer in numerator form only (using negative exponents where necessary).

$\frac{\left(4 x y^{-2}\right)\left(-12 x^{-4} y^{12}\right)}{6 x^{2} y}$

[3]

$\left(2 x^{-1} y^{-2}\right)^{-3}\left(4 x^{2} y^{3}\right)^{4}$

[3]

Question 4

HLPaper 1

Simplify the following expressions.

$\sqrt{\left(9 x^{6} y^{-2} z^{4}\right)^{3}}(3 x y z)^{-2}$

[3]

$\left(\frac{x^{2} z^{4}}{2 y^{5}}\right)\left(\frac{3 x^{2} y^{3}}{z^{2}}\right)^{2}$

[3]

Question 5

HLPaper 2

Consider the expression $x^{0.75} \cdot y^{1.25}$ .

Write the expression $\log(x^{0.75} \cdot y^{1.25})$ in terms of $\log(x)$ and $\log(y)$ .

[2]

If $x = 16$ and $y = 81$ , evaluate $x^{0.75} \cdot y^{1.25}$ .

[2]

Question 6

HLPaper 2

Find the value of $k$ .

[2]

Calculate the volume after 5 minutes.

[2]

Determine the rate of change of volume with respect to time when $t = 3$ .

[3]

Question 7

HLPaper 1

Consider the expression $a^{\frac{1}{5}} \times a^{\frac{2}{5}}$ , which can be written in the form $a^{p}$ for $a > 0$ .

Find the value of $p$ .

[2]

Given that $a^{\frac{1}{5}} \times a^{\frac{2}{5}} = 8$ , find the value of $a$ .

[2]

Question 8

HLPaper 2

A biologist is studying the growth of a particular bacteria population in a lab. The population of the bacteria, $P(t)$ , at time $t$ (in hours) is modeled by the function $P(t) = 100e^{0.3t}$ .

Find the population of the bacteria after 5 hours.

[2]

Determine the time it takes for the population to double.

[2]

Question 9

HLPaper 2

Express the following in the form $x^{r}$

$(\sqrt[5]{x})^{6}$

[2]

$\sqrt[4]{\sqrt[3]{x}}$

[2]

$\frac{1}{\sqrt[3]{x^{4}}}$

[2]

Question 10

HLPaper 2

A scientist is studying the growth of a bacterial culture. The population $P$ (in thousands) after $t$ hours can be modeled by the equation $P(t) = 2.5t^{1.4} + 3$ .

Find the population after 2 hours.

[2]

Calculate the time when the population reaches 20 thousand bacteria.

[3]

Paper

Level

Question 1

HLPaper 1

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.

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

[2]

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

[3]

Question 2

HLPaper 1

Questions on algebra and surds.

Solve the equation $2-x \sqrt{3}=\frac{7 x}{\sqrt{3}}$ , giving your answer in the form $\frac{\sqrt{a}}{b}$ , where $a, b \in \mathbb{Z}$ . Find the value of $a$ and the value of $b$ .

[3]

Given that $8 \sqrt{2}=2^{k}$ , find the value of $k$ .

[2]

Question 3

HLPaper 1

Simplify the following expressions. Express the final answer in numerator form only (using negative exponents where necessary).

$\frac{\left(4 x y^{-2}\right)\left(-12 x^{-4} y^{12}\right)}{6 x^{2} y}$

[3]

$\left(2 x^{-1} y^{-2}\right)^{-3}\left(4 x^{2} y^{3}\right)^{4}$

[3]

Question 4

HLPaper 1

Simplify the following expressions.

$\sqrt{\left(9 x^{6} y^{-2} z^{4}\right)^{3}}(3 x y z)^{-2}$

[3]

$\left(\frac{x^{2} z^{4}}{2 y^{5}}\right)\left(\frac{3 x^{2} y^{3}}{z^{2}}\right)^{2}$

[3]

Question 5

HLPaper 2

Consider the expression $x^{0.75} \cdot y^{1.25}$ .

Write the expression $\log(x^{0.75} \cdot y^{1.25})$ in terms of $\log(x)$ and $\log(y)$ .

[2]

If $x = 16$ and $y = 81$ , evaluate $x^{0.75} \cdot y^{1.25}$ .

[2]

Question 6

HLPaper 2

Find the value of $k$ .

[2]

Calculate the volume after 5 minutes.

[2]

Determine the rate of change of volume with respect to time when $t = 3$ .

[3]

Question 7

HLPaper 1

Consider the expression $a^{\frac{1}{5}} \times a^{\frac{2}{5}}$ , which can be written in the form $a^{p}$ for $a > 0$ .

Find the value of $p$ .

[2]

Given that $a^{\frac{1}{5}} \times a^{\frac{2}{5}} = 8$ , find the value of $a$ .

[2]

Question 8

HLPaper 2

A biologist is studying the growth of a particular bacteria population in a lab. The population of the bacteria, $P(t)$ , at time $t$ (in hours) is modeled by the function $P(t) = 100e^{0.3t}$ .

Find the population of the bacteria after 5 hours.

[2]

Determine the time it takes for the population to double.

[2]

Question 9

HLPaper 2

Express the following in the form $x^{r}$

$(\sqrt[5]{x})^{6}$

[2]

$\sqrt[4]{\sqrt[3]{x}}$

[2]

$\frac{1}{\sqrt[3]{x^{4}}}$

[2]

Question 10

HLPaper 2

A scientist is studying the growth of a bacterial culture. The population $P$ (in thousands) after $t$ hours can be modeled by the equation $P(t) = 2.5t^{1.4} + 3$ .

Find the population after 2 hours.

[2]

Calculate the time when the population reaches 20 thousand bacteria.

[3]

AHL 1.10—Expressions with non-integer exponents Questionbank