3 Marks Question

Question 13 Marks

Simplify and express in exponential form: $\left[\left(5^{2}\right)^{3} \times 5^{4}\right] \div 5^{7}$

Answer

In the above question,
We have to simplify the given numbers into exponential form:
$\left[\left(5^2\right)^3 \times 5^4\right] \div 5^7$
Using identity: $\left.\left(a^m\right)^n=a^{m n}\right)$
$=\left[(5)^2 \times 3 \times 5^4\right] \div 5^7$
$=\left[(5)^6 \times 4\right] \div 5^7$
Using identity: ( $a^m \times a^n=a^{m+n}$ )
$=\left[5^{6+4}\right] \div 5^7$
Using identity: ( $\left.a^m \div a^n=a^{m-n}\right)$
Therefore,
$=5^{10} \div 5^7$
$=5^{10-7}=5^3$

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Question 23 Marks

Simplify and express in exponential form: $\frac{2^{3} \times 3^{4} \times 4}{3 \times 32}$

Answer

In the above question,
We have to simplify the given numbers into exponential form:
Therefore, We have,
$\frac{2^{2} \times 3^{4} \times 4}{3 \times 32}=\frac{2^{3} \times 3^{4} \times 2 \times 2}{3 \times 2 \times 2 \times 2 \times 2 \times 2}$ = $\frac{2^{3} \times 3^{4} \times 2^{2}}{3 \times 2^{5}}$ = $\frac{2^{3} \times 2^{2} \times 3^{4}}{3 \times 2^{5}}$
$=\frac{2^{5} \times 3^{4}}{3 \times 2^{5}}\left(a^{m} \times a^{n}=a^{m+n}\right)$
Using identity: $\left(a^m \div a^n=a^{m-n}\right)$
$=2^{5-5} \times 3^{4-1}$
$=2^0 3^3$
$=1 \times 3^3$
$=3^3$

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Question 33 Marks

Work out $(1)^5,(-1)^3,(-1)^4,(-10)^3,(-5)^4$.

Answer

i. We have $(1)^5=1 \times 1 \times 1 \times 1 \times 1=1$
In fact, you will realise that 1 raised to any power is 1 .
ii. $(-1)^3=(-1) \times(-1) \times(-1)=1 \times(-1)=-1$
iii. $(-1)^4=(-1) \times(-1) \times(-1) \times(-1)=1 \times 1=1$
You may check that $(-1)$ raised to any odd power is $(-1)$, and $(-1)$ raised to any even power is $(+1)$.
iv. $(-10)^3=(-10) \times(-10) \times(-10)=100 \times(-10)=-1000$
v. $(-5)^4=(-5) \times(-5) \times(-5) \times(-5)=25 \times 25=625$

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