1 Marks Question

Question 11 Mark

If $a = 3$ and $b = -2,$ find the values of : $a + b^{ab}$

Answer

We have,
$a^b + b^a$
$= (3 + (-2))^{3(-2)}$
$= (3 − 2)^{-6}$
$= (1)^{−6} $
$= 1$

Question 21 Mark

If a = 3 and b = -2, find the values of:
$\text{a}^\text{a}+\text{b}^\text{b}$

Answer

We have,
$\text{a}^\text{a}+\text{b}^\text{b}$
$=3^3+(-2)^{-2}$
$=3^3+(-\frac{1}{2})^2$
$=27+\frac{1}{4}$
$=\frac{109}{4}$

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Question 31 Mark

If $493992 = a^4b^2c^3,$ find the values of $a, b$ and $c$ are and are different positive primes.

Answer

Taking out the $\text{LCM,}$ the factors are $2^4, 3^2$ and $7^3 a^4 b^2 c^3 = 2^4, 3^2$
and $7^3 a = 2, b = 3$
and $c = 7 [$since, $a b$ and $c$ are primes$].$

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Question 41 Mark

Simplify the following : $3(a^4 b^3)^{10} \times 5(a^2 b^2)^3$

Answer

$3(a^4 b^3)^{10} \times 5(a^2 b^2)^3$
$= 3(a^{40} b^{30}) \times 5(a^6 b^6)$
$= 15(a^{46} b^{36})$

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Question 51 Mark

Prove that:
$\Big(\frac{\text{x}^\text{a}}{\text{x}^\text{b}}\Big)^\text{c}\times\Big(\frac{\text{x}^\text{b}}{\text{x}^\text{c}}\Big)^\text{a}\times\Big(\frac{\text{x}^\text{c}}{\text{x}^\text{a}}\Big)^\text{b}=1$

Answer

To prove,
$\Big(\frac{\text{x}^\text{a}}{\text{x}^\text{b}}\Big)^\text{c}\times\Big(\frac{\text{x}^\text{b}}{\text{x}^\text{c}}\Big)^\text{a}\times\Big(\frac{\text{x}^\text{c}}{\text{x}^\text{a}}\Big)^\text{b}=1$
Left hand side (LHS) = Right hand side (RHS) Considering LHS,
$=\Big(\frac{\text{x}^{\text{ac}}}{\text{x}^{\text{bc}}}\Big)\times\Big(\frac{\text{x}^{\text{ba}}}{\text{x}^{\text{ca}}}\Big)\times\Big(\frac{\text{x}^{\text{bc}}}{\text{x}^{\text{ab}}}\Big)$
$=\text{x}^{\text{ac}-\text{bc}}\times\text{x}^{\text{ba}-\text{ca}}\times\text{x}^{\text{bc}-\text{ab}}$
$=\text{x}^{\text{ac}-\text{bc}+\text{ba}-\text{ca}+\text{bc}-\text{ab}}$
$=\text{x}^0$
$=1$
Therefore, LHS = RHS
Hence proved.

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Question 61 Mark

Simplify the following : $(2x^{-2} y^3)^3$

Answer

$(2x^{-2} y^3)^3$
$(2^3 \times x^{-2 \times 3} y^{3 \times 3}) $
$= 8x^{-6}y^9$

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Question 71 Mark

If a = 3 and b = -2, find the values of:
$\text{a}^\text{b}+\text{b}^\text{a}$

Answer

We have,$\text{a}^\text{b}+\text{b}^\text{a}$
$=3^{-2}+(-2)^{3}$
$=\big(\frac{1}{3}\big)^2+(-2)^3$
$=\frac{1}{9}+-8$
$=-\frac{71}{9}$

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Question 81 Mark

Given $4725=3^\text{a}\ 5^\text{b}\ 7^\text{c}$ find,The value of $2^{-\text{a}}3^\text{b}7^\text{c}$

Answer

The value of $2^{-\text{a}}\times3^\text{b}\times7^\text{c}$
Sol:
$2^{-a}\times3^\text{b}\times7^\text{c}=2^{-3}\times3^2\times7^1$
$2^{-3}\times3^2\times7^1=\frac{1}{8}\times9\times7$
$\frac{63}{8}$

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Question 91 Mark

Given $4725=3^\text{a}\ 5^\text{b}\ 7^\text{c}$ find,The integral values of a, b and c

Answer

Taking out the LCM of 4725, we get
$3^3\times5^2\times7^1=3^\text{a}\times5^\text{b}\times7^\text{c}$
By comparing, we get
$\text{a}=\text{3},\ \text{b}=2$ and $\text{c}=1.$

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