Indices

Simplify.

a × a × a × a × a × a × a

4b × 5b²

Solve.

6 k - 3 = 177

Complete each statement by writing the missing power in the box.

(4³)² = 4

Work out.

2^-4

Write 5 × 5 × 5 × 5 × 5 × 5 as a power of 5.

Write down the value of 1¹⁰.

Work out.

4 × 4 × 4 - 3³

Find the value of k.

A teacher asks their class which of the following values is the smallest.

2^-1 4^-3 10^-2

One student says

2^-1 is the smallest because 2 is smaller than 4 and 10.

Explain the error made by the student in their reasoning and write down the smallest value.

........ The smallest value is........

Find the value of:

2⁵,

Simplify.

y¹² ÷ y⁴

Find the value of p.

The table shows some numbers each written as a power of 4.

Complete the table to show 16 and 64 each written as a power of 2.

A number is written as 4²⁰.

Use a pattern in the table to help you write this number as a power of 2.

Work out.

1₃¹ 8

2^x × 4^y = 16

Write the following in order of size, smallest first.

0.2 2^-2 2 × 10^-2

Show how you decide.

smallest

Complete each statement by writing the missing power in the box.

6³ × 6⁴ = 6

2²⁴ = 2

Simplify.

h × h × h × h

Factorise fully.

3 fg + 12 g

Write the following in order of size, smallest first. 0.2 2^-2 2 × 10^-2

Show how you decide.

smallest

Find the value of m.

Solve.

Work out.

Work out.

3⁵

Find the value of y.

384 = 6 × 4^y

Write 3^–1 as a fraction.

Show that the value of x is

The kth term of a sequence is r^k, where r ≠ 0. The sixth term is equal to three times the second term.

Find the value of r, giving your answer correct to 3 decimal places.

Simplify.

5 r - 7 t - 3 r + 2 t

a × a × a × a × a

7b⁵ ÷ b

Factorise.

4 a - 12 b

Complete each statement by writing the missing power in the box.

6 × 6 × 6 = 6

16 = 2

Work out.

A student says that they have placed the following values in order starting with the smallest.

Has the student done this correctly? Show how you decide.

A student says that they have placed the following values in order starting with the smallest.

Has the student done this correctly? Show how you decide.

Given that (2^k)⁶ × 8 = 2⁴⁵, find the value of k.

Complete this statement by writing the missing power in the box.

For each statement, complete the box to show the power of 10.

One million = 10

One thousandth = 10

Reece is given this question.

Write 20 as a product of prime factors. Give your answer in index form.

Reece’s answer is 2 × 2 × 5.

Is Reece correct? Explain your answer.

Complete the power of 2.

Work out.

Simplify.

x¹² ÷ x⁴

Write 3 × 3 × 3 × 3 as a power of 3.

Show that the answer to 2⁶ × 4^-1 is a square number.

Simplify.

(x⁴)³

Complete the power of 2 for each statement by writing the missing value in the box.

2 × 2^y = 1.

Find the value of y.

Round 356 to the nearest ten.

Round 356.052 to 1 decimal place.

Find the value of y in each of the following.

3 × 3 × 3 × 3 = 3^y

6³ × 6⁵ = 6^y

Simplify.

h³ × h^-3

The length of each side of a plastic cube is 2 a millimetres. The cube has mass 32a² grams.

Find an expression for the density of the cube in its simplest form. Give the units of your answer.

You are given that

Find the lowest common multiple (LCM) of 270 and 177 147. Give your answer using power notation and as an ordinary number.

(a)(i) using power notation........

as an ordinary number........

Write 177 147 000 000 as a product of its prime factors.

3ⁿ = 177147 × 9⁵.

Find the value of n.

Find the value of x in each of the following.

a⁴ × a³ = a^x

(b⁴)³ = b^x

Factorise fully.

18x² + 9x

Work out 7².

Write 6^–1 as a fraction.

Work out 2⁴.

Find the value of n.

Two numbers, P and Q, are written as products of their prime factors.

Find the lowest common multiple (LCM) of P and Q.

The number C is written as the product of its prime factors.

Work out P ÷ C, leaving your answer as a product of powers of prime numbers.

Write 450 as a product of its prime factors.

Find the highest common factor (HCF) of 270 and 450.

b is a vector. Given that b + is parallel to , find two possible answers for b.

Given that

find the value of m and the value of n.

Simplify.

5 x – 6 y – x + 3 y

w⁸ ÷ w²

5c²d × 3c

Work out the value of

4 x – 7 when x = 5,

Show that a⁵ × (a³)² can be expressed as a¹¹.

Given that y¹⁸ ÷ y⁶ = y^k, find the value of k.

In the following equation, n is an integer greater than 1.

Find k when n = 7.

Find n when k = 64.

Show that can be written in the form a + b

Show that a⁵ × (a³)² can be expressed as a¹¹.

Write 7 × 7 × 7 × 7 as a power of 7.

Complete this working to write 4³ as a power of 2.

4³ = 4 × 4........

so 4³ = 2 × 2 × 2 ×........

Write these numbers in order, starting with the largest.

8.1 × 10¹ 1.02 × 10³ 9.83 × 10^–2 3 × 10²

largest

Simplify.

a⁶ ÷ a²

(b⁵)³

Factorise.

Evaluate.

4^-2

Work out.

Fill in the power.

5 = 125

Find the value of y.

Find the values of z.

(b) z =........ or z =........