DB125: Maths for Diploma Studies: Exponents and Exponentiation Reinforcement Questions

Exponents and Exponentiation Reinforcement Questions

What is the answer to the calculation below?

$begin mathsize 14px style 22 to the power of negative 4 end exponent space plus space 14 comma 000 comma 000 to the power of 1 space plus space 128 to the power of 0 end style$

With a negative exponent, the result is always a fraction with 1 as the numerator and the denominator being the original number raised by the power of the exponent, so:

$begin mathsize 14px style 22 to the power of negative 4 end exponent space space i s space space 1 over 22 to the power of 4 space equals space 4.27 space cross times space 10 to the power of negative 6 end exponent end style$

A number raised to the first power is the original number so:

$begin mathsize 14px style 14 comma 000 comma 000 to the power of 1 space equals space 14 comma 000 comma 000 end style$

And a number raised to the power of 0 is always 1. Altogether, this gives us:

$begin mathsize 14px style 4.27 space cross times space 10 to the power of negative 6 end exponent space plus space 14 comma 000 comma 000 space plus space 1 space equals space 14 comma 000 comma 001 end style$

The impact of the 22^-4 is only seen if we show the answer out to six decimal places.

What is the sum of $begin mathsize 14px style a space plus space b end style$ from the calculations below?

$begin mathsize 14px style a space equals space open parentheses 12 space cross times space 3 close parentheses cubed end style$

$begin mathsize 14px style b space equals space 7 to the power of 4 space cross times space 7 cubed end style$

We can start by calculating $begin mathsize 14px style a end style$ :

$begin mathsize 14px style a space equals space left parenthesis 12 space cross times space 3 right parenthesis cubed space equals space 12 to the power of 3 end exponent space cross times space 3 cubed end style$

$begin mathsize 14px style a space equals space left parenthesis 12 space cross times space 3 right parenthesis cubed space equals space 1 comma 728 space cross times space 27 end style$

$begin mathsize 14px style a space equals space 1 comma 728 space cross times space 27 space equals space 46 comma 656 end style$

And now $begin mathsize 14px style b end style$ :

$begin mathsize 14px style b space equals space 7 to the power of 4 space cross times space 7 cubed space equals space 7 to the power of 7 end style$

$begin mathsize 14px style b space equals space 7 to the power of 7 space equals space 823 comma 543 end style$

And finally $begin mathsize 14px style a space plus space b end style$ :

$begin mathsize 14px style a space plus space b space equals space 46 comma 656 space plus space 823 comma 543 space equals space 870 comma 199 end style$

What do we get if we subtract $begin mathsize 14px style a end style$ from $begin mathsize 14px style b end style$ in the calculations below?

$begin mathsize 14px style a space equals space open parentheses 124 over 14 close parentheses to the power of 5 end style$

$begin mathsize 14px style b space equals space open parentheses 18 squared close parentheses cubed end style$

Let’s start with calculating $begin mathsize 14px style a end style$ :

$begin mathsize 14px style a space equals space open parentheses 124 over 14 close parentheses to the power of 5 end style$

$begin mathsize 14px style a space equals space 124 to the power of 5 over 14 to the power of 5 space end style$

$begin mathsize 14px style a space equals space fraction numerator 29 comma 316 comma 250 comma 624 over denominator 537 comma 824 end fraction end style$

$begin mathsize 14px style a space equals space 54 comma 509 end style$

And now $begin mathsize 14px style b end style$ :

$begin mathsize 14px style b space equals space left parenthesis 18 squared right parenthesis cubed end style$

$begin mathsize 14px style b space equals space 18 to the power of open parentheses 2 space cross times space 3 close parentheses end exponent end style$

$begin mathsize 14px style b space equals space 18 to the power of 6 end style$

$begin mathsize 14px style b space equals space 34 comma 012 comma 224 end style$

And now subtract $begin mathsize 14px style a end style$ from $begin mathsize 14px style b end style$ :

$begin mathsize 14px style b space minus space a space equals space 34 comma 012 comma 224 space minus space 54 comma 509 space equals space 33 comma 957 comma 715 end style$

Calculate the following.

$begin mathsize 14px style 18 cubed over 21 to the power of negative 2 end exponent end style$

Here the exponents stay associated with the bases, so it is a simple calculation:

$begin mathsize 14px style 18 cubed over 21 to the power of negative 2 end exponent space equals space fraction numerator 5 comma 832 over denominator 0.00227 end fraction space equals space 2 comma 571 comma 912 end style$

What is $begin mathsize 14px style a end style$ to the power $begin mathsize 14px style b end style$ from the calculations below?

$begin mathsize 14px style a space equals space 5 to the power of 4 divided by 5 to the power of 5 end style$

$begin mathsize 14px style b space equals space 2 squared space cross times space 2 to the power of 4 end style$

Let’s start with calculating $begin mathsize 14px style a end style$ :

$begin mathsize 14px style a space equals space 5 to the power of 4 divided by 5 to the power of 5 space equals space 5 to the power of left parenthesis 4 space minus space 5 right parenthesis end exponent end style$

$begin mathsize 14px style a space equals space 5 to the power of left parenthesis 4 space minus space 5 right parenthesis end exponent space equals space 5 to the power of negative 1 end exponent end style$

$begin mathsize 14px style a space equals space 5 to the power of negative 1 end exponent space equals space 1 over 5 to the power of 1 end style$

$begin mathsize 14px style a space equals space 1 fifth space equals space 0.2 end style$

And now $begin mathsize 14px style b end style$ :

$begin mathsize 14px style b space equals space 2 squared space cross times space 2 to the power of 4 end style$

$begin mathsize 14px style b space equals space 2 to the power of left parenthesis 2 space plus space 4 right parenthesis end exponent space equals space 2 to the power of 6 end style$

$begin mathsize 14px style b space equals space 2 to the power of 6 space equals space 64 end style$

And now we work out what $begin mathsize 14px style a end style$ to the power $begin mathsize 14px style b end style$ is:

$begin mathsize 14px style a to the power of b space equals space 0.2 to the power of 64 space equals space 1.84 space cross times space 10 to the power of negative 45 end exponent end style$

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Maths for Diploma Studies

Exponents and Exponentiation Reinforcement Questions

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