Question 1

LEVEL 6

Simplify (10x)^{-2}

 

Select the correct answer from the list below:

A: \dfrac{1}{100x^2}

B: 100x^2

C: 100x^{-2}

D: \dfrac{1}{100x^{-2}}

 

CORRECT ANSWER: A: \dfrac{1}{100x^2}

WORKED SOLUTION:

Remember your indices rule here,

the negative power turns the expression into a fraction, so

\dfrac{1}{{(10x)}^2}

Then you have to perform the power rule where the

power 2 applies to both the number and x value giving

\dfrac{1}{100x^2}

Question 2

LEVEL 6

Write 8^7\times 16^5 in the form 2^x, where x is an integer.

You cannot use a calculator for this question.

 

Select the correct answer from the list below:

A: 2^{72}

B: 2^{41}

C: 2^{19}

D: 2^{25}

 

CORRECT ANSWER: B: 2^{41}

WORKED SOLUTION:

Before we can use any of our laws of indices we need to recognise that 16 and 8 can be written as powers of 2.

16=2^4

 

8=2^3

Therefore,

8^7\times 16^5=(2^3)^7\times(2^4)^5

Using the power law, we get

(2^3)^7\times(2^4)^5=2^{(3\times7)}\times2^ {(4\times5)}=2^{21}\times2^{20}

And finally, we can use our multiplication law to add the powers

2^{21}\times2^{20}=2^{21+20}=2^{41}

Hence,

8^7\times 16^5=2^{41}

Question 3

LEVEL 6

Write 25^7\times 4^8 in standard form.

You cannot use a calculator for this question.

 

Select the correct answer from the list below:

A: 5\times10^{7}

B: 2.5\times10^{11}

C: 1.25\times10^{16}

D: 4\times10^{14}

 

CORRECT ANSWER: D: 4\times10^{14}

WORKED SOLUTION:

To get our answer in standard form, we need to have it as \times10^x, where x is an integer.

To start, we need to realise that we can to make a 10, we can multiply 5 and 2, which we can get by changing how we write 25 and 4.

25=5^2

 

4=2^2

Therefore,

25^7\times 4^8 =(5^2)^7\times(2^2)^8

Using the power law, we get

(5^2)^7\times(2^2)^8=2^{(2\times7)}\times2^ {(2\times8)}=5^{14}\times2^{16}

And now, we need to make our \times10 by pulling out pairs of 2 and 5.

5^{14}\times2^{16}=5\times5^{13}\times2\times2^{15}

 

5\times5^{13}\times2\times2^{15}=5\times2\times5^{13}\times2^{15}

 

5\times2\times5^{13}\times2^{15} =10\times5^{13}\times2^{15}

 

10\times5^{13}\times2^{15}=10\times5\times5^{12}\times2\times2^{14}

 

10\times5\times5^{12}\times2\times2^{14} =10\times5\times2\times5^{12}\times2^{14}

 

10\times5\times2\times5^{12}\times2^{14} =10\times10\times5^{12}\times2^{14}

 

10\times10\times5^{12}\times2^{14}=10^2\times5^{12}\times2^{14}

 

We can do this as many times as the lowest power, i.e. 14 times, removing 1 from the power each time. So, we will end up with:

5^{14}\times2^{16} =10^{14}\times2^2

Now, we’re nearly there. Firstly, we need to remember that we have the power of 10 at the end

5^{14}\times2^{16} =2^2\times10^{14}

And finally, we need to write the 2^2=4

5^{14}\times2^{16} =4\times10^{14}

Question 4

LEVEL 6

Simplify 8^{\frac{2}{3}}

 

Select the correct answer from the list below:

A: 4

B: 8

C: 8^2

D: 64

 

CORRECT ANSWER: A: 4

WORKED SOLUTION:

Fractional powers need to be completed in two steps.

Firstly, the denominator tells you what root to perform, in this case cube root.

We know that \sqrt[3]8 = 2

Secondly, you use the numerator to tell you what power to perform, in this case 2

2^2 = 4

Question 5

LEVEL 6

Simplify \dfrac{4^4 \times 4^{-2}}{4^2}

 

Select the correct answer from the list below:

A: 2

B: 4^-2

C: 1

D: 4

 

CORRECT ANSWER: C: 1

WORKED SOLUTION:

Simplify the fraction by cancelling down to get

4^2 \times 4^{-2}

Then use the indices rule that a negative power can be written as a fraction

4^2 \times\dfrac{1}{4^2}

 

Then simplify to

\dfrac{4^2}{4^2}

 

which cancels down to

1