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Question 511 Mark

Classify the following number as rational or irrational. give reasons to support your answer.
$\sqrt{21}$

Answer

$\sqrt{21}=\sqrt{3}\times\sqrt{7}=4.58257...$
It is an irrational number.

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Question 521 Mark

Find an irrational number between 5 and 6.

Answer

An irrational number between 5 and 6 $=\sqrt{5\times6}=\sqrt{30}$

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Question 531 Mark

Simplify:
$3^\frac{1}{4}\times5^\frac{1}{4}$

Answer

$3^\frac{1}{4}\times5^\frac{1}{4}=(3\times5)^{\frac{1}{4}}=(15)^{\frac{1}{4}}$

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Question 541 Mark

Represent the following rational numbers on the number line:
$1.3$

Answer

$1.3=\frac{13}{10}=1\frac{3}{10}$

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Question 551 Mark

Classify the following number as rational or irrational. give reasons to support your answer.
$\sqrt{\frac{3}{81}}$

Answer

$\sqrt{\frac{3}{81}}$
$\sqrt{\frac{3}{81}}=\sqrt{\frac{1}{27}}=\frac{1}{3}\sqrt{\frac{1}{3}}$
It is an irrational number.

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Question 561 Mark

Simplify:
$\frac{5^{\frac{6}{7}}}{5^{\frac{2}{3}}}$

Answer

$\frac{5^{\frac{6}{7}}}{5^{\frac{2}{3}}}=5^{\big(\frac{6}{7}-\frac{2}{3}\big)}$
$=5^{\big(\frac{18-14}{21}\big)}=5^{\frac{4}{21}}$

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Question 571 Mark

Write the rationalising factor of the denominator in $\frac{1}{\sqrt{2}+\sqrt{3}}.$

Answer

The rationalising factor of the denominator in $\frac{1}{\sqrt{2}+\sqrt{3}}$ is $\big(\sqrt{2}-\sqrt{3}\big).$

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Question 581 Mark

Without actual division, find the following rational numbers are terminating decimals. $\frac{16}{125}$

Answer

$\frac{16}{125}$ Denominator of $\frac{16}{125}$ is $125.$
And,
$125 = 5^3$ Therefore, $125$ has no other factors than $2$ and $5.$
Thus, $\frac{16}{125}$ is a terminating decimal.

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Question 591 Mark

Represent the following rational numbers on the number line:
$-2.4$

Answer

$-2.4=\frac{-24}{10}=\frac{-12}{5}=-2\frac{2}{5}$

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Question 601 Mark

Find two irrational numbers between 0.16 and 0.17.

Answer

Two irrational numbers between 0.16 and 0.17 are as follows: 0.1611161111611111611111... and 0.169669666...

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Question 611 Mark

Classify the following number as rational or irrational. give reasons to support your answer.
2.356565656...

Answer

2.356565656... is a rational number because it is repeating.

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Question 621 Mark

Simplify:
$2^\frac{2}{3}\times2^\frac{1}{3}$

Answer

$2^{\frac{2}{3}}\times2^{\frac{1}{3}}$
$=2^{\frac{2}{3}+\frac{1}{3}}$
$=2^{\frac{3}{3}}$
$=2^1$
$=2$

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Question 631 Mark

Evaluate:
$\big(64\big)^{\frac{1}{6}}$

Answer

$\big(64\big)^{\frac{1}{6}}=(2^6)^{\frac{1}{6}}=2^{\big(6\times\frac{1}{6}\big)}=2^1=2$

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Question 641 Mark

Represent the following rational numbers on the number line:
$-\frac{23}{6}$

Answer

$-\frac{23}{6}=-3\frac{5}{6}$

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Question 651 Mark

Give an example of two irrational numbers whose:
Quotient is a rational number.

Answer

2 irrational numbers with quotient a rational number will be $\sqrt{63}$ and $\sqrt{7}$

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Question 661 Mark

Simplify:
$7^\frac{5}{6}\times7^\frac{2}{3}$

Answer

$\Bigg(7^{\frac{5}{6}}\times7^{\frac{2}{3}}\Bigg)=7^{\big(\frac{5}{6}+\frac{2}{3}\big)}=7^{\big(\frac{5+4}{6}\big)}$
$=7^{\frac{9}{6}}=7^{\frac{3}{2}}$

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Question 671 Mark

Give an example of two irrational numbers whose:
Difference is an irrational number.

Answer

2 irrational numbers with difference an irrational number will be $3-\sqrt{5}$ and $3+\sqrt{5}.$

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Question 681 Mark

Evaluate:
$\big(81\big)^{\frac{3}{4}}$

Answer

$\big(81\big)^{\frac{3}{4}}=(3^4)^{\frac{3}{4}}=3^{\big(4\times\frac{3}{4}\big)}=3^3=27$

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Question 691 Mark