Powers, Roots & Standard Form (OCR GCSE Maths: Higher): Exam Questions

One sheet of paper is  $9\times {10}^{-3}$cm thick.

Mark wants to put $500$sheets of paper into the paper tray of his printer.
The paper tray is $4$ cm deep.

Is the paper tray deep enough for $500$ sheets of paper?
You must explain your answer.

Write the following numbers in order of size. Start with the smallest number.

Write down the value of ${10}^0$.

Write down the value of ${10}^{-2}$.

Write these numbers in order of size.
Start with the smallest number.

Write down the value of  ${100}^{\frac{1}{2}}$.

Find the value of ${125}^{\frac{2}{3}}$.

Write down the value of ${36}^{\frac{1}{2}}$.

Write down the value of ${23}^0$.

Work out the value of ${27}^{-\frac{2}{3}}$.

The table shows some information about eight planets.

Write down the name of the planet with the greatest mass.

Find the difference between the mass of Venus and the mass of Mercury.

Nishat says that Neptune is over a hundred times further away from Earth than Venus is.

Is Nishat right?
You must show how you get your answer.

Work out $(13.8\times {10}^7)\times (5.4\times {10}^{-12})$
Give your answer as an ordinary number.

Write down the value of 10⁰.

Write 6.7 x 10^-5 as an ordinary number.

Work out the value of (3 x 10⁷) x (9 x 10⁶).

Give your answer in standard form.

Calculate $9\times {10}^4\times 3\times {10}^3$.

Give your answer in standard form.

Work out the value of $(9\times {10}^{-4})\times (3\times {10}^7)$.
Give your answer in standard form.

Work out the value of 25^-3.

Work out the value of 350³.
Give your answer in standard form.

Patrick has to work out the exact value of ${64}^{\frac{1}{4}}$

Patrick says,

      "$\frac{1}{4}$ of $64$ is $16$ so ${64}^{\frac{1}{4}}=16$"

Explain what is wrong with what Patrick says.

Write ${\left(\sqrt[4]{8}\right)}^5$ as a power of 2.

You are given that $177147=3^{11}$

$3^n=177147\times 9^5$

Find the value of $n.$

Work out.

$\sqrt[3]{64}\times 2^{-1}$

$4.3\times {10}^5+3.8\times {10}^4$

Give your answer in standard form.

Carol says that ${64}^{-\frac{1}{2}}=\frac{1}{32}$

Explain her error and give the correct value of ${64}^{-\frac{1}{2}}$in the form $\frac{p}{q}$.

A company makes sweets. The sweets are put into packets.

Here are some facts.

Calculate the mean number of sweets in one packet.

Sweets are made on 288 days each year.

Calculate the number of sweets made each year. Give your answer in standard form.

The company has 152 machines making the sweets. Each machine operates for 15 hours each day.

i) Calculate the number of sweets made by one machine each hour. Give your answer as an ordinary number correct to the nearest 10.

[3]

ii) State one assumption you have made in part (c)(i).

[1]

The table shows information about the surface area of each of the world’s oceans.

Work out the difference, in square kilometres, between the surface area of the Atlantic Ocean and the surface area of the Indian Ocean.
Give your answer in standard form.

.......................square kilometres

The surface area of the Pacific Ocean is $k$ times the surface area of the Arctic Ocean.

Work out the value of $k$.
Give your answer correct to the nearest whole number.

$k=$...........................

The Sun has been burning for $4.6\times {10}^9$ years.

Scientists estimate that the Sun will continue to burn for another $5\times {10}^8$ years before it becomes a red giant star.

Explain why it is appropriate to use standard form for these numbers.

Find the total time, from when the Sun started burning to when it becomes a red giant.

Give your answer in standard form.

A grain of sand has a mass of approximately $6.4\times {10}^{-5}$ grams.

A bag of sand has a total mass of 5 kg.

Use these values to estimate the number of grains of sand in the bag.

Give your answer in standard form, correct to 2 significant figures.

You are given that

$5^{k+1}\times 25=5^7$ and ${\left(3^m\right)}^4=3^{12}$

Find the value of $k$ and the value of $m$.

Find the value of $2^{-3}$.

$5\sqrt{5}$ can be written in the form $5^k$.

Find the value of $k$.

Find the value of ${81}^{-\frac{1}{2}}$.

Find the value of ${\left(\frac{64}{125}\right)}^{\frac{2}{3}}$.

Write down the value of ${64}^{\frac{1}{2}}$.

Find the value of ${\left(\frac{8}{125}\right)}^{-\frac{2}{3}}$.

Find the value of $\sqrt[3]{8\times {10}^6}$.

Find the value of     ${144}^{\frac{1}{2}}\times {64}^{-\frac{1}{3}}$.

Solve     $3^{2x}=\frac{1}{81}$.

Write $640000000$ in standard form.

Work out $(3\times {10}^7)÷(6\times {10}^4)$
Give your answer in standard form.

Write $5400000$ as a number in standard form.

Write $3.2\times {10}^{-4}$as an ordinary number.

The mass of the Sun is $2\times {10}^{30}$ kg.
The mass of the largest known star is $315$times the mass of the Sun.

Work out the mass of this star.
Give your answer in kg in standard form.

Write $7.97\times {10}^{-6}$ as an ordinary number.

Work out the value of $(2.52\times {10}^5)÷(4\times {10}^{-3})$
Give your answer in standard form.

      $p^2=\frac{x-y}{xy}$

$$\begin{gathered} x=8.5\times {10}^9 \\ y=4\times {10}^8 \end{gathered}$$

Find the value of $p$.
Give your answer in standard form correct to $2$ significant figures.

$T=\sqrt{\frac{w}{d^3}}$

$$\begin{gathered} w=5.6\times {10}^{-5} \\ d=1.4\times {10}^{-4} \end{gathered}$$

Work out the value of $T$.
Give your answer in standard form correct to $3$ significant figures.

$w$ is increased by $10\%$
$d$ is increased by $5\%$

Lottie says,          
"The value of $T$ will increase because both $w$ and $d$ are increased."

Lottie is wrong.
Explain why.

Write 8.2 x 10⁵ as an ordinary number.

Write 0.000 376 in standard form.

Work out the value of (2.3 x 10¹²) ÷ (4.6 x 10³) Give your answer in standard form.

Work out the value of (3.5 x  10⁶) ÷ (5 x 10^-3).
Give your answer in standard form.

There are two errors in Sam’s method for finding the value of ${64}^{\frac{{}^{-}2}{3}}$shown below.

Find the cube root of 64 and then multiply by 2.
The cube root of 64 is 4 and then 4 x 2 = 8.
The negative power makes the answer negative so answer equals –8.

Describe these errors and then give the correct value of ${64}^{\frac{{}^{-}2}{3}}$.

Correct value ........................................................

Show that  $\frac{\sqrt[3]{81}}{3}$ can be written as $3^{\frac{1}{3}}$

Simplify  $8^2\times \sqrt[3]{4^6}$

Give your answer in the form 2^a where $a$ is an integer.

Show each stage of your working clearly.

$a=25\times {10}^{14n}$ where $n$is an integer.

Find an expression, in terms of $n$ , for $a^{\frac{3}{2}}$

Give your answer in standard form.