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Question 511 Mark

If $a = 1$, $b = 2$ then find the value of $(\text{a}^{\text{b}}+\text{b}^{\text{a}})^{-1}$.

Answer

Given, $a = 1$ and $b = 2$
$\therefore(\text{a}^{\text{b}}+\text{b}^{\text{a}})^{-1}$
$=\frac{1}{\text{a}^{\text{b}}+\text{b}^{\text{a}}}$
$=\frac{1}{1^2+2^1}$
$=\frac{1}{1+2}$
$=\frac{1}{3}$

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Question 521 Mark

Simplify $6\sqrt{36}+5\sqrt{12}$

Answer

$6\sqrt{3}+5\sqrt{12}$ $=6\sqrt{3}+5\sqrt{4\times3}$ $=6\sqrt{3}+5\times2\sqrt{3}$ $=6\sqrt{3}+10\sqrt{3}$ $=16\sqrt{3}$

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Question 531 Mark

Simplify $\big(2\sqrt{5}+3\sqrt{2}\big)^2.$

Answer

$\big(2\sqrt{5}+3\sqrt{2}\big)^2$ $=\big(2\sqrt{5}\big)^2+2\times2\sqrt{5}\times3\sqrt{2}+\big(3\sqrt{2}\big)^2$ $=20+12\sqrt{10}+18$ $=38+12\sqrt{10}$

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Question 541 Mark

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

Answer

$2$ irrational numbers with product a rational number will be $5+\sqrt{7}$ and $5-\sqrt{7}$

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Question 551 Mark

Classify the following number as rational or irrational. give reasons to support your answer. $\sqrt{361}$

Answer

$\sqrt{361}=19$ So, it is rational.

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Question 561 Mark

Evaluate: $\big(25\big)^{\frac{3}{2}}$

Answer

$\big(25\big)^{\frac{3}{2}}=(5^2)^{\frac{3}{2}}=5^{\big(2\times\frac{3}{2}\big)}=5^3=125$

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Question 571 Mark

Simplify: $\Big(3^{\frac{1}{3}}\Big)^4$

Answer

$\Big(3^{\frac{1}{3}}\Big)^4=3^{\big(\frac{1}{3}\times4\big)}=3^{\frac{4}{3}}$

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Question 581 Mark

Simplify: $\Big({\frac{1}{3^4}}\Big)^{\frac{1}{2}}$

Answer

$\Big({\frac{1}{3^4}}\Big)^{\frac{1}{2}}=\big(3^{-4}\big)^{\frac{1}{2}}=3^{\big(-4\times\frac{1}{2}\big)}=3^{-2}$

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Question 591 Mark

If $\sqrt{10}=3.162,$ find the value of $\frac{1}{\sqrt{10}}.$

Answer

Given, $\sqrt{10}=3.162$
Now,
$\frac{1}{\sqrt{10}}=\frac{1}{\sqrt{10}}\times\frac{\sqrt{10}}{\sqrt{10}}=\frac{\sqrt{10}}{\big(\sqrt{10}\big)^2}=\frac{\sqrt{10}}{10}=\frac{3.162}{100}=0.3162$

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Question 601 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 611 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 621 Mark

Evaluate: $\big(8\big)^{-\frac{1}{3}}$

Answer

$\big(8\big)^{-\frac{1}{3}}=\frac{1}{\big(8\big)^{\frac{1}{3}}}=\frac{1}{\big(2^3\big)^{\frac{1}{3}}}=\frac{1}{2^{\big(3\times\frac{1}{3}\big)}}$ $=\frac{1}{2^1}=\frac{1}{2}$

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Question 631 Mark

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

Answer

$2$ irrational numbers with difference is a rational number will be $5+\sqrt{3}$ and $2+\sqrt{3}$

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Question 641 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 651 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 661 Mark

Find the value of $\frac{21\sqrt{12}}{10\sqrt{27}}.$

Answer

$\frac{21\sqrt{12}}{10\sqrt{27}}$ $=\frac{21\sqrt{4\times3}}{10\sqrt{9\times3}}$ $=\frac{21\times2\sqrt{3}}{10\times3\sqrt{3}}$ $=\frac{7}{5}$

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Question 671 Mark

Simplify: $2^\frac{5}{8}\times3^\frac{5}{8}$

Answer

$2^\frac{5}{8}\times3^\frac{5}{8}=(2\times3)^{\frac{5}{8}}=(6)^{\frac{5}{8}}$

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Question 681 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 691 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 701 Mark

Let $x$ be a rational number and $y$ be an irrational number. Is $x+y$ necessarily an irrational number? Give a example in support of your answer.

Answer

$x$ be a rational number and $y$ be an irrational number then $x+y$ necessarily will be an irrational number. Example: 5 is a rational number but $\sqrt{2}$ is irrational. So, $5+\sqrt{2}$ will be an irrational number.

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Question 711 Mark

Evaluate: $(1^3+2^3+3^3)^{\frac{1}{2}}$

Answer

$(1^3+2^3+3^3)^{\frac{1}{2}}$ $=(1+8+27)^{\frac{1}{2}}$ $=(36)^{\frac{1}{2}}$ $=(6^2)^{\frac{1}{2}}$ $=6$

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1 Marks Question - Page 2 - Maths STD 9 Questions - Vidyadip

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