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Hint: The given problem is related to estimation of product by rounding off. Round off the given numbers to nearest hundreds and estimate the product.

Complete step-by-step answer:

Before proceeding with the solution, we must understand the concept of rounding off a number.

If the first non-zero digit from the left of a number is less than $5$, then the digit to its immediate left is left untouched and the digit used for reference is changed to zero. Similarly, if the first non-zero digit from the left of a number is greater than or equal to $5$, then the digit to its immediate left is increased by $1$ and the digit used for reference is changed to zero.

For example: Letâ€™s consider the number \[3251\] . Here, the first non-zero digit from the left is $1$. Since $1$ is smaller than $5$ , the digit to its immediate left, i.e. $5$ is left untouched and the reference digit, i.e. $1$ is changed to zero. So, the number, on rounding off, becomes $3250$ . Again, consider the number $3256$ . Here, the first non-zero digit from the left is $6$. Since $6$ is greater than $5$ , the digit to its immediate left, i.e. $5$ is increased by $1$ and the reference digit, i.e. $6$ is changed to zero. So, the number, on rounding off, becomes $3260$ .

Now, coming to the question, we are asked to estimate the value of $578\times 161$. The first number is $578$ . On rounding off to the nearest tens, we get $580$. We will further round it off to the nearest hundreds. So, on rounding off $580$ to nearest hundreds , we get $600$ . The second number is $161$ . On rounding off to the nearest tens, we get $160$. We will further round it off to the nearest hundreds. So, on rounding off $160$ to the nearest hundreds, we get $200$.

So, $578\times 161$ will be approximately equal to $600\times 200=120000$.

Hence, the estimated value of $578\times 161$ is $120000$.

Note: On finding products by rounding off, the value obtained will be approximately equal to the actual value and it will not be exactly equal to the actual value. This method should only be used if approximate answers are required. If exact values are required, then this method will not be fruitful.

Complete step-by-step answer:

Before proceeding with the solution, we must understand the concept of rounding off a number.

If the first non-zero digit from the left of a number is less than $5$, then the digit to its immediate left is left untouched and the digit used for reference is changed to zero. Similarly, if the first non-zero digit from the left of a number is greater than or equal to $5$, then the digit to its immediate left is increased by $1$ and the digit used for reference is changed to zero.

For example: Letâ€™s consider the number \[3251\] . Here, the first non-zero digit from the left is $1$. Since $1$ is smaller than $5$ , the digit to its immediate left, i.e. $5$ is left untouched and the reference digit, i.e. $1$ is changed to zero. So, the number, on rounding off, becomes $3250$ . Again, consider the number $3256$ . Here, the first non-zero digit from the left is $6$. Since $6$ is greater than $5$ , the digit to its immediate left, i.e. $5$ is increased by $1$ and the reference digit, i.e. $6$ is changed to zero. So, the number, on rounding off, becomes $3260$ .

Now, coming to the question, we are asked to estimate the value of $578\times 161$. The first number is $578$ . On rounding off to the nearest tens, we get $580$. We will further round it off to the nearest hundreds. So, on rounding off $580$ to nearest hundreds , we get $600$ . The second number is $161$ . On rounding off to the nearest tens, we get $160$. We will further round it off to the nearest hundreds. So, on rounding off $160$ to the nearest hundreds, we get $200$.

So, $578\times 161$ will be approximately equal to $600\times 200=120000$.

Hence, the estimated value of $578\times 161$ is $120000$.

Note: On finding products by rounding off, the value obtained will be approximately equal to the actual value and it will not be exactly equal to the actual value. This method should only be used if approximate answers are required. If exact values are required, then this method will not be fruitful.

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