Value: Understanding Multiplication Equivalence

The problem asks us to identify which of the given options is numerically equal to the expression $7362 \times 726.8 \times 0.709$. This requires understanding how moving the decimal point in a number affects its value and how these changes can be balanced within a multiplication.

Multiplication: Decimal Shifts Explained

To find an equivalent expression, we can manipulate the decimal places of the numbers involved. When we move the decimal point in a number:

• Moving the decimal point to the left by one place is the same as dividing the number by 10 (or multiplying by $10^{-1}$).
• Moving the decimal point to the right by one place is the same as multiplying the number by 10 (or multiplying by $10^{1}$).

We need to compare the original numbers with the numbers in the options. Let's examine the fourth option: $73.62 \times 7268 \times 7.09$.

We can express each number in this option in terms of the original numbers:

• $73.62$ vs $7362$: The decimal point in $73.62$ has been moved two places to the left compared to $7362$. This means $73.62 = 7362 \times 10^{-2}$.
• $7268$ vs $726.8$: The decimal point in $7268$ has been moved one place to the right compared to $726.8$. This means $7268 = 726.8 \times 10^{1}$.
• $7.09$ vs $0.709$: The decimal point in $7.09$ has been moved one place to the right compared to $0.709$. This means $7.09 = 0.709 \times 10^{1}$.

Equal Value Calculation Steps

Now, let's substitute these relationships back into the fourth option's expression:

$73.62 \times 7268 \times 7.09$
$= (7362 \times 10^{-2}) \times (726.8 \times 10^{1}) \times (0.709 \times 10^{1})$

We can rearrange the terms using the commutative and associative properties of multiplication:

$= (7362 \times 726.8 \times 0.709) \times (10^{-2} \times 10^{1} \times 10^{1})$

Now, we multiply the powers of 10 using the rule $a^m \times a^n = a^{m+n}$:

$= (7362 \times 726.8 \times 0.709) \times 10^{-2+1+1}$
$= (7362 \times 726.8 \times 0.709) \times 10^{0}$

Since any number raised to the power of 0 is 1 ($10^0 = 1$), the expression becomes:

$= (7362 \times 726.8 \times 0.709) \times 1$
$= 7362 \times 726.8 \times 0.709$

Final Confirmation of Equal Value

The calculation shows that the expression $73.62 \times 7268 \times 7.09$ is indeed equal to the original expression $7362 \times 726.8 \times 0.709$. The shifts in the decimal points were compensated by multiplying by powers of 10 that ultimately resulted in a factor of $10^0$, confirming the equivalence.