Find the values of below
(i) Additive inverse of $\dfrac{-7}{19}$
(ii) Multiplicative inverse of -13
(iii) Probability of getting a head in a toss of a coin
(iv) Square root of 64
(v) $n\times n\times n\times n\times n$ = ………………..
(vi) Area of circle = ……………….
(vii) Volume of a cube = ………………
(viii) 1m = ………… cm.
(ix) Value of ${{2}^{-3}}$is = …………..
(x) Factorize: $5xy+10x$ = ……………
Answer
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Hint: For the question in the sub-part (i) We solve this question going through the definition of additive inverse and using the formula for the additive inverse, $b$ is an additive inverse of $a$ if $a+b=0$.
Complete step by step answer:
For sub-part (ii) We solve this question going through the definition of multiplicative inverse and using the formula for the multiplicative inverse, $b$ is the multiplicative inverse of $a$ if $ab=1$.
For sub-part (iii) We solve this question by going through the definition of the probability and use the formula for the probability of getting an event A $P\left( A \right)=\dfrac{n}{N}$, where n is occurrences of event A and total possible outcomes are N.
For sub-part (iv) We solve this question by going through the definition of the square root and use the formula for it as $y=\sqrt{x}\Rightarrow {{y}^{2}}=x$.
For sub-part (v) We solve this question by going through the definition of the product of any number r times and use the formula for it as $x\times x\times .....\times x\left( r\text{ times} \right)={{x}^{r}}$.
For sub-part (vi) We solve this question by first considering the formula for the area of the circle $\text{Area of Circle =}\pi \times {{\left( \text{radius} \right)}^{2}}$ and find the required answer.
For sub-part (vii) We solve this question by considering the formula for volume of cube, $\text{Volume of cube =}{{\left( \text{side} \right)}^{3}}$ and use it to find the required answer.
For sub-part (viii) We solve this question by considering the formula for conversion of meters into centimeters, I meter is equal to 100 centimeters and find the answer.
For sub-part (ix) We solve this question by using the formula ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ and then multiply the number 2 three times and find the answer.
For sub-part (x) We solve this question by factorizing the given equation by taking 5x common from 5xy and 10x.
(i) Additive inverse of $\dfrac{-7}{19}$
Let us consider the definition of the Additive Inverse.
The additive inverse of a number is a real number that sums up to the given number and gives zero.
Let ‘a’ be a real number and ‘b’ be its additive inverse.
Then, by the definition of additive inverse, we get
$\Rightarrow \text{ }a+b=0$
We consider the given number $\dfrac{7}{19}$ as $a$.
Let us consider the additive inverse as $b$.
Now, we need to find b.
Then, by using the definition of additive inverse stated above, we get
$\begin{align}
& \Rightarrow \text{ }a+b=0 \\
& \Rightarrow \text{ }\dfrac{-7}{19}+b=0 \\
& \Rightarrow \text{ }b=-\dfrac{-7}{19} \\
& \Rightarrow \text{ }b=\dfrac{7}{19} \\
\end{align}$
Therefore, the additive inverse of the given number $\dfrac{-7}{19}$ is $\dfrac{7}{19}$.
Hence, answer is $\dfrac{7}{19}$.
(ii) Multiplicative inverse of -13
The multiplicative inverse of a number is nothing but the reciprocal of the number or Multiplicative inverse of a number is a number whose product with a given number gives one. So, multiplicative inverse of $x$ is $\dfrac{1}{x}$.
Let a be a real number and b be its multiplicative inverse.
Then, by the definition of multiplicative inverse, we get
$\begin{align}
& \Rightarrow \text{ }ab=1 \\
& \Rightarrow \text{ }b=\dfrac{1}{a} \\
\end{align}$
Now, let us consider the given number -13 as $a$.
We need to find the number $b$.
From the definition of multiplicative inverse, we get
$\begin{align}
& \Rightarrow \text{ }b=\dfrac{1}{a} \\
& \Rightarrow \text{ }b=\dfrac{1}{-13} \\
& \Rightarrow \text{ }b=-\dfrac{1}{13} \\
\end{align}$
Therefore, the multiplicative inverse of -13 is $-\dfrac{1}{13}$.
(iii) Probability of getting a head in a toss of a coin.
Let us start by considering the definition of Probability.
If a random experiment can result in N number of cases and n of them are favorable to the occurrence of event A, then the probability of occurrence of A is given by,
$P\left( A \right)=\dfrac{n}{N}$
Now, let us consider the event of getting head as event A.
The total number of outcomes while tossing a coin is 2, as we can get 1 Head and 1 Tail when a coin is tossed. So, we get N=2.
Here, the number of cases favorable to event A is only 1, as we can get only one head when a coin is tossed. Here we get n=1.
Then, from the definition, we get
$P\left( A \right)=\dfrac{1}{2}$
Therefore, the probability of getting head in a toss of a coin is $\dfrac{1}{2}$.
Hence answer is $\dfrac{1}{2}$.
(iv) Square root of 64
Let x and y as two real numbers, then the square root of x is y if
$\begin{align}
& \Rightarrow {{y}^{2}}=x \\
& \Rightarrow y=\sqrt{x} \\
\end{align}$
Now, we consider x = 64, then
$\begin{align}
& \Rightarrow {{y}^{2}}=64 \\
& \Rightarrow {{y}^{2}}={{8}^{2}} \\
& \Rightarrow y=8 \\
\end{align}$
Therefore, the square root of 64 is 8.
Hence answer is 8.
(v) $n\times n\times n\times n\times n$
We solve this problem by multiplying a number n, 5 times to itself.
When a number $x$ is multiplied $r$ times by itself, it gives ${{x}^{r}}$.
So, multiplying $n$ five times by itself gives ${{n}^{5}}$.
Therefore, $n\times n\times n\times n\times n={{n}^{5}}$.
Hence answer is ${{n}^{5}}$.
(vi) Area of circle = ……………….
Let us use the formula for the area of the circle.
$\text{Area of Circle =}\pi \times {{\left( \text{radius} \right)}^{2}}$
So, for any circle with radius r units, the area is $\pi \times {{r}^{2}}$ square units.
Hence answer is $\pi \times {{r}^{2}}$.
(vii) Volume of cube = ……………….
Let us use the formula for the area of the circle.
$\text{Volume of cube =}{{\left( \text{side} \right)}^{3}}$
So, for any cube with side $a$units, volume is ${{a}^{3}}$ square units.
Hence answer is ${{a}^{3}}$.
(viii) 1m = ……..
As 1 centimeter is equal to 100 centimeters,
So, 1m=100cm
Hence the answer is 100cms.
(ix) Value of ${{2}^{-3}}$ is
For any number a, ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$
Therefore, we get
$\begin{align}
& \Rightarrow {{2}^{-3}}=\dfrac{1}{{{2}^{3}}} \\
& \Rightarrow {{2}^{-3}}=\dfrac{1}{8} \\
\end{align}$
Hence answer is $\dfrac{1}{8}$.
(x) Factorize 5xy + 10x
In factorization of any expression, we need to check if we have any common terms in the algebraic expression.
Here in 5xy and 10x, we can see that x is common in both and 10 is divisible by 5. So, by taking them common we get
5xy + 10x = 5x(y+2)
Hence answer is 5x(y+2).
Note:
For subpart (i) Here in this question one might a mistake by writing $-\left( \dfrac{-7}{19} \right)$ as $\dfrac{-7}{19}$. But need to remember that there are two minus signs in it and the minus of minus is a plus. So, the answer will be $\dfrac{7}{19}$.
For subpart (ii) One should be careful while applying the definitions and should not confuse the definition of multiplicative inverse with the additive inverse.
For subpart (iii) In this question, one might make a mistake by taking the formula for probability given above as $P\left( A \right)=\dfrac{N}{n}$. But it wrong as N should be in the denominator.
For subpart (iv) There is a possibility of making a mistake while finding the square root by writing the answer as -8 and 8. But as the square root of any number should be positive, we should take only 8 as the answer.
For subpart (v) There is a possibility of making a mistake while solving by writing the answer as 5n as there is 5 numbers of n’s in the question. But it as wrong as we get 5n while adding them not while multiplying.
For subpart (vi) The common mistake made while solving this question is one might confuse the formula for the area of a circle as $Area=2\pi r$, but it is the formula for the perimeter of the circle.
For subpart (vii) There is a possibility of making a mistake by taking the formula for the volume of the cube as $Volume=6\times {{\left( side \right)}^{2}}$, but is the formula for the area of cube not for volume.
For subpart (viii) One might make a mistake by confusing the formula 1 meter = 100 centimeters with 1 centimeter = 10 millimeters and use the formula as 1 meter = 10 centimeters.
For sub part (ix) One might make a mistake in this question by taking the formula ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ as ${{a}^{-n}}=-{{a}^{n}}$. So, one need to remember these formulas carefully.
For subpart (x) There is a possibility of making a mistake by taking only x as common and write the answer as x(5y+10). But we need to take all the possible factors common and write the answer as 5x(y+2).
Complete step by step answer:
For sub-part (ii) We solve this question going through the definition of multiplicative inverse and using the formula for the multiplicative inverse, $b$ is the multiplicative inverse of $a$ if $ab=1$.
For sub-part (iii) We solve this question by going through the definition of the probability and use the formula for the probability of getting an event A $P\left( A \right)=\dfrac{n}{N}$, where n is occurrences of event A and total possible outcomes are N.
For sub-part (iv) We solve this question by going through the definition of the square root and use the formula for it as $y=\sqrt{x}\Rightarrow {{y}^{2}}=x$.
For sub-part (v) We solve this question by going through the definition of the product of any number r times and use the formula for it as $x\times x\times .....\times x\left( r\text{ times} \right)={{x}^{r}}$.
For sub-part (vi) We solve this question by first considering the formula for the area of the circle $\text{Area of Circle =}\pi \times {{\left( \text{radius} \right)}^{2}}$ and find the required answer.
For sub-part (vii) We solve this question by considering the formula for volume of cube, $\text{Volume of cube =}{{\left( \text{side} \right)}^{3}}$ and use it to find the required answer.
For sub-part (viii) We solve this question by considering the formula for conversion of meters into centimeters, I meter is equal to 100 centimeters and find the answer.
For sub-part (ix) We solve this question by using the formula ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ and then multiply the number 2 three times and find the answer.
For sub-part (x) We solve this question by factorizing the given equation by taking 5x common from 5xy and 10x.
(i) Additive inverse of $\dfrac{-7}{19}$
Let us consider the definition of the Additive Inverse.
The additive inverse of a number is a real number that sums up to the given number and gives zero.
Let ‘a’ be a real number and ‘b’ be its additive inverse.
Then, by the definition of additive inverse, we get
$\Rightarrow \text{ }a+b=0$
We consider the given number $\dfrac{7}{19}$ as $a$.
Let us consider the additive inverse as $b$.
Now, we need to find b.
Then, by using the definition of additive inverse stated above, we get
$\begin{align}
& \Rightarrow \text{ }a+b=0 \\
& \Rightarrow \text{ }\dfrac{-7}{19}+b=0 \\
& \Rightarrow \text{ }b=-\dfrac{-7}{19} \\
& \Rightarrow \text{ }b=\dfrac{7}{19} \\
\end{align}$
Therefore, the additive inverse of the given number $\dfrac{-7}{19}$ is $\dfrac{7}{19}$.
Hence, answer is $\dfrac{7}{19}$.
(ii) Multiplicative inverse of -13
The multiplicative inverse of a number is nothing but the reciprocal of the number or Multiplicative inverse of a number is a number whose product with a given number gives one. So, multiplicative inverse of $x$ is $\dfrac{1}{x}$.
Let a be a real number and b be its multiplicative inverse.
Then, by the definition of multiplicative inverse, we get
$\begin{align}
& \Rightarrow \text{ }ab=1 \\
& \Rightarrow \text{ }b=\dfrac{1}{a} \\
\end{align}$
Now, let us consider the given number -13 as $a$.
We need to find the number $b$.
From the definition of multiplicative inverse, we get
$\begin{align}
& \Rightarrow \text{ }b=\dfrac{1}{a} \\
& \Rightarrow \text{ }b=\dfrac{1}{-13} \\
& \Rightarrow \text{ }b=-\dfrac{1}{13} \\
\end{align}$
Therefore, the multiplicative inverse of -13 is $-\dfrac{1}{13}$.
(iii) Probability of getting a head in a toss of a coin.
Let us start by considering the definition of Probability.
If a random experiment can result in N number of cases and n of them are favorable to the occurrence of event A, then the probability of occurrence of A is given by,
$P\left( A \right)=\dfrac{n}{N}$
Now, let us consider the event of getting head as event A.
The total number of outcomes while tossing a coin is 2, as we can get 1 Head and 1 Tail when a coin is tossed. So, we get N=2.
Here, the number of cases favorable to event A is only 1, as we can get only one head when a coin is tossed. Here we get n=1.
Then, from the definition, we get
$P\left( A \right)=\dfrac{1}{2}$
Therefore, the probability of getting head in a toss of a coin is $\dfrac{1}{2}$.
Hence answer is $\dfrac{1}{2}$.
(iv) Square root of 64
Let x and y as two real numbers, then the square root of x is y if
$\begin{align}
& \Rightarrow {{y}^{2}}=x \\
& \Rightarrow y=\sqrt{x} \\
\end{align}$
Now, we consider x = 64, then
$\begin{align}
& \Rightarrow {{y}^{2}}=64 \\
& \Rightarrow {{y}^{2}}={{8}^{2}} \\
& \Rightarrow y=8 \\
\end{align}$
Therefore, the square root of 64 is 8.
Hence answer is 8.
(v) $n\times n\times n\times n\times n$
We solve this problem by multiplying a number n, 5 times to itself.
When a number $x$ is multiplied $r$ times by itself, it gives ${{x}^{r}}$.
So, multiplying $n$ five times by itself gives ${{n}^{5}}$.
Therefore, $n\times n\times n\times n\times n={{n}^{5}}$.
Hence answer is ${{n}^{5}}$.
(vi) Area of circle = ……………….
Let us use the formula for the area of the circle.
$\text{Area of Circle =}\pi \times {{\left( \text{radius} \right)}^{2}}$
So, for any circle with radius r units, the area is $\pi \times {{r}^{2}}$ square units.
Hence answer is $\pi \times {{r}^{2}}$.
(vii) Volume of cube = ……………….
Let us use the formula for the area of the circle.
$\text{Volume of cube =}{{\left( \text{side} \right)}^{3}}$
So, for any cube with side $a$units, volume is ${{a}^{3}}$ square units.
Hence answer is ${{a}^{3}}$.
(viii) 1m = ……..
As 1 centimeter is equal to 100 centimeters,
So, 1m=100cm
Hence the answer is 100cms.
(ix) Value of ${{2}^{-3}}$ is
For any number a, ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$
Therefore, we get
$\begin{align}
& \Rightarrow {{2}^{-3}}=\dfrac{1}{{{2}^{3}}} \\
& \Rightarrow {{2}^{-3}}=\dfrac{1}{8} \\
\end{align}$
Hence answer is $\dfrac{1}{8}$.
(x) Factorize 5xy + 10x
In factorization of any expression, we need to check if we have any common terms in the algebraic expression.
Here in 5xy and 10x, we can see that x is common in both and 10 is divisible by 5. So, by taking them common we get
5xy + 10x = 5x(y+2)
Hence answer is 5x(y+2).
Note:
For subpart (i) Here in this question one might a mistake by writing $-\left( \dfrac{-7}{19} \right)$ as $\dfrac{-7}{19}$. But need to remember that there are two minus signs in it and the minus of minus is a plus. So, the answer will be $\dfrac{7}{19}$.
For subpart (ii) One should be careful while applying the definitions and should not confuse the definition of multiplicative inverse with the additive inverse.
For subpart (iii) In this question, one might make a mistake by taking the formula for probability given above as $P\left( A \right)=\dfrac{N}{n}$. But it wrong as N should be in the denominator.
For subpart (iv) There is a possibility of making a mistake while finding the square root by writing the answer as -8 and 8. But as the square root of any number should be positive, we should take only 8 as the answer.
For subpart (v) There is a possibility of making a mistake while solving by writing the answer as 5n as there is 5 numbers of n’s in the question. But it as wrong as we get 5n while adding them not while multiplying.
For subpart (vi) The common mistake made while solving this question is one might confuse the formula for the area of a circle as $Area=2\pi r$, but it is the formula for the perimeter of the circle.
For subpart (vii) There is a possibility of making a mistake by taking the formula for the volume of the cube as $Volume=6\times {{\left( side \right)}^{2}}$, but is the formula for the area of cube not for volume.
For subpart (viii) One might make a mistake by confusing the formula 1 meter = 100 centimeters with 1 centimeter = 10 millimeters and use the formula as 1 meter = 10 centimeters.
For sub part (ix) One might make a mistake in this question by taking the formula ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ as ${{a}^{-n}}=-{{a}^{n}}$. So, one need to remember these formulas carefully.
For subpart (x) There is a possibility of making a mistake by taking only x as common and write the answer as x(5y+10). But we need to take all the possible factors common and write the answer as 5x(y+2).
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