Question 158(i)

Find a single machine that will do the same job as the given hook-up.
$a (\times 2^{3}) m a c h i n e f o l l o w e d b y (\times 2^{- 2}) m a c h i n e$ .
Maya multiplied $(4^{2} \times 3^{2})$ by thinking about stretching machines.

Use Maya's idea to multiply $5^{3} \times 2^{3}$ .
Maya's idea is another product law of exponents.
Multiplying Expressions with the same Exponents
$a^{c} \times b^{c} = (a \times b)^{c}$
You can use this law with more than two expressions. If the exponents are the same, multiply the expressions by multiplying the bases and using the same exponent. e.g. $2^{8} \times 3^{8} \times 7^{8} = (2 \times 3 \times 7)^{8} = 42^{8}$ .

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Solution

$(\times 2^{3})$ machine followed by $(\times 2^{- 2})$ machine. So, it produces $2 \times 2 \times 2 \times \frac{1}{2} \times \frac{1}{2} = 2^{8}$
Hence, $(\times 2^{1})$ single machine can do the same job as the given hook-up.

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Question 158(iii)

Find a single machine that will do the same job as the given hook-up.
$a (\times 5^{99})$ machine followed by a $(5^{- 100})$ machine.
Maya multiplied $(4^{2} \times 3^{2})$ by thinking about stretching machines.

Use Maya's idea to multiply $5^{3} \times 2^{3}$ .
Maya's idea is another product law of exponents.
Multiplying Expressions with the same Exponents
$a^{c} \times b^{c} = (a \times b)^{c}$
You can use this law with more than two expressions. If the exponents are the same, multiply the expressions by multiplying the bases and using the same exponent. e.g. $2^{8} \times 3^{8} \times 7^{8} = (2 \times 3 \times 7)^{8} = 42^{8}$ .

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 153(ii)

On multiplying terms with the same base, the exponents are ____.

Q. Product Law of Exponents :-
In multiplication, if base is the same then add exponents.
Which of the following can be solved using Product law of exponents?

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(×23) machine followed by (×2−2) machine. So, it produces 2×2×2×12×12=28 Hence,(×21) single machine can do the same job as the given hook-up.

$(\times 2^{3})$ machine followed by $(\times 2^{- 2})$ machine. So, it produces $2 \times 2 \times 2 \times \frac{1}{2} \times \frac{1}{2} = 2^{8}$
Hence, $(\times 2^{1})$ single machine can do the same job as the given hook-up.