Question 153(i)

In a repeater machine with 0 as an exponent, the base machine is applied 0 times.
What do these machines do to a piece of chalk?

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Solution

Since, $3^{0} = 1, 13^{0} = 1, 29^{0} = 1$
Using law of exponents, $a^{0} = 1$ [ $∵$ a is non-zero interger]
So, machine $(\times 3^{0}), (\times 13^{0}) a n d (\times 29^{0})$ produce nothing on not change the piece 7 chalk.

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Question 151

Find three repeater machines that will do the same work as a $(\times 64)$ machine. Draw them. or describe them using exponents.

Question 151

Find three repeater machines that will do the same work as a $(\times 64)$ machine. Draw them. or describe them using exponents.

Question 153(ii)

In a repeater machine with 0 as an exponent, the base machine is applied 0 times.
What do you think the value of $6^{o}$ is ?
You have seen that a hook up of repeater machines with the same base can be replaced by a single repeater machine. Similarly, when you multiply exponential expressions with the same base, you can replace them with a single expression.
Asif Raza thought about how he could rewrite the expression $2^{2} 0 \times 2^{5}$ .

Asif Raza's idea is one of the product laws of exponents, which can be expressed like this
Multiplying Expressions with the same base $a^{b} \times a^{c} = a^{b + c}$
Actually, this law can be used with more than two expressions. As long as the bases are the same, to find the product you can add the exponents and use the same base.
e.g. $3^{2} \times 3^{3} \times 3^{10} = 3^{2 + 3 + 10} = 3^{15}$

Question 152

What will the following machine do to a 2 cm long piece of chalk ?

Question 149

Repeater Machine
Similarly, repeater machine is a hypothetical machine which automatically enlarges items several times. e.g. Sending a piece of wire through a $(\times 2^{4})$ machine is the same as putting it through a $(\times 2)$ machine four times.
So, if you send a 3 cm piece of wire thorugh a $(\times 2)^{4}$ machine, its length becomes $3 \times 2 \times 2 \times 2 \times 2 = 48 c m .$ It can also be written that a base (2) machine is being applied 4 times.

What will be the new length of a 4 cm strip inserted in the machine?

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Question 153(i) In a repeater machine with 0 as an exponent, the base machine is applied 0 times. What do these machines do to a piece of chalk?

Since, 30=1,130=1,290=1 Using law of exponents, a0=1 [∵ a is non-zero interger] So, machine (× 30),(× 130) and (× 290) produce nothing on not change the piece 7 chalk.

Question 153(i)

In a repeater machine with 0 as an exponent, the base machine is applied 0 times.
What do these machines do to a piece of chalk?

Since, $3^{0} = 1, 13^{0} = 1, 29^{0} = 1$
Using law of exponents, $a^{0} = 1$ [ $∵$ a is non-zero interger]
So, machine $(\times 3^{0}), (\times 13^{0}) a n d (\times 29^{0})$ produce nothing on not change the piece 7 chalk.