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Gr 12 Exam Feb 2017 Paper 1 Question 1.1.2

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\text{1.1 Solve for } x
\quad\text{1.1.2 } \quad\sqrt{x^3}=512

The question is allocated three marks, and thus you have 3 minutes to solve this, but you may not need the full three minutes if you find the right approach.

SOLUTION:

\sqrt{x^3}=512

x^3=512*512

Note that 512*512 is a large number, and you should be wary of trying to calculate such a large number during the exam, also considering that there are only three marks given. You can easily calculate this using your calculator, but will this provide a step in the solution? You could end up doing too much work when there are simpler approaches.

You may not be aware that there is a factorise function on your calculator, which provides the prime factors of integers. Factorisation is available on the Casio fx-82ZA PLUS,as the FACT function, and also on the Sharp EL-W535HT as the P.FACT function. Both will display 2^9 when this function is used with 512 as input. Using this our problem is translated to the following.

x^3=2^9*2^9

For this problem, which is exponent-based, it is better it to see the number 512 as a power of 2, so that you can then using exponent mathematics. You should have memorised this as part of your mental arithmetic knowledge. If not, then take the time to memorise these and print out the page with the mental arithmetic powers, and keep this with you while you are doing other things. These numbers occur frequently in problems found in examinations, and you do not want to be delayed when you need to break down a number into its factors. As an example, what is 81 a power of? (Answer at the end of this page).

x^3=2^{18}

This is one jump using one of the exponential rules.

x^3=2^{6*3}

This step could be omitted, since it is only showing the examiner that you know that 18 can be factored as 6*3

x^3=(2^{6})^3

This is important, since both sides now have the same exponent, and since both sides are equal, this then means that the bases must be equal.

Thus

x=2^6

x=64

This seems like quite a lot of work for solving this problem and for the three marks, but not all of these steps will be required, and you can omit some if they appear to be sufficiently obvious.

RETURN TO 81

Consider 81. You should immediately see that this is divisible by 3 (since 8+1=9 which is itself divisible by 3). Dividing by 3 gives 27, which is also divisible by 3, giving 9, which is equal to 3*3.

So 81=3^4

TRY

Solve for x where:

\sqrt[3]{x^4}

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