Question

Evaluate the following: \[{\left( {10.4} \right)^3}\].
 (a) \[1124.846\]
 (b) \[1124.864\]
 (c) \[1124.964\]
 (d) \[1124.874\]

Answer 1

Hint: Here, we need to find the value of the given cube \[{\left( {10.4} \right)^3}\]. We will write the given number as the cube of a sum of two numbers. Then, we will use the algebraic identity for the cube of the sum of two numbers and simplify the expression to get the required value.

Formula Used: The cube of the sum of two numbers \[a\] and \[b\] is given by the algebraic identity \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\].

Complete step-by-step answer:
First, we will write the number \[10.4\] as the sum of two numbers.
Rewriting the number \[10.4\], we get
\[10.4 = 10 + 0.4\]
Substituting \[10.4 = 10 + 0.4\] in the given cube \[{\left( {10.4} \right)^3}\], we get
\[ \Rightarrow {\left( {10.4} \right)^3} = {\left( {10 + 0.4} \right)^3}\]
Now, we know that \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\].
Therefore, substituting \[a = 10\] and \[b = 0.4\] in the identity, we get
\[ \Rightarrow {\left( {10 + 0.4} \right)^3} = {10^3} + {\left( {0.4} \right)^3} + 3\left( {10} \right)\left( {0.4} \right)\left( {10 + 0.4} \right)\]
or
\[ \Rightarrow {\left( {10.4} \right)^3} = {10^3} + {\left( {0.4} \right)^3} + 3\left( {10} \right)\left( {0.4} \right)\left( {10 + 0.4} \right)\]
We will simplify the expression on the right hand side to get the required answer.
Applying the exponents on the bases, we get
\[ \Rightarrow {\left( {10.4} \right)^3} = 1000 + 0.064 + 3\left( {10} \right)\left( {0.4} \right)\left( {10 + 0.4} \right)\]
Multiplying the terms 3, 10, and \[0.4\], we get
\[ \Rightarrow {\left( {10.4} \right)^3} = 1000 + 0.064 + 12\left( {10 + 0.4} \right)\]
Multiplying the terms using the distributive law of multiplication, we get
\[ \Rightarrow {\left( {10.4} \right)^3} = 1000 + 0.064 + 120 + 4.8\]
Now, adding the terms in the equation, we get
\[ \Rightarrow {\left( {10.4} \right)^3} = 1124.864\]
\[\therefore\] We get the exact value of \[{\left( {10.4} \right)^3}\] as \[1124.864\].
Thus, the correct option is option (b).

Note: We have used the distributive property of multiplication to find the product of 12 and \[\left( {10 + 0.4} \right)\]. The distributive property of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].
We can also use the algebraic identity for the cube of the difference of two numbers to evaluate \[{\left( {10.4} \right)^3}\].
The cube of the difference of two numbers \[a\] and \[b\] is given by the algebraic identity \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)\].
Substituting \[a = 11\] and \[b = 0.6\] in the identity, we get
\[ \Rightarrow {\left( {11 - 0.6} \right)^3} = {11^3} - {\left( {0.6} \right)^3} - 3\left( {11} \right)\left( {0.6} \right)\left( {11 - 0.6} \right)\]
Therefore, we get
\[ \Rightarrow {\left( {10.4} \right)^3} = {11^3} - {\left( {0.6} \right)^3} - 3\left( {11} \right)\left( {0.6} \right)\left( {11 - 0.6} \right)\]
We will simplify the expression on the right hand side to get the required answer.
Applying the exponents on the bases, we get
\[ \Rightarrow {\left( {10.4} \right)^3} = 1331 - 0.216 - 3\left( {11} \right)\left( {0.6} \right)\left( {11 - 0.6} \right)\]
Multiplying the terms 3, 11, and \[0.6\], we get
\[ \Rightarrow {\left( {10.4} \right)^3} = 1331 - 0.216 - 19.8\left( {11 - 0.6} \right)\]
Multiplying the terms using the distributive law of multiplication, we get
\[ \Rightarrow {\left( {10.4} \right)^3} = 1331 - 0.216 - 217.8 + 11.88\]
Now, adding and subtracting the terms in the equation, we get
\[ \Rightarrow {\left( {10.4} \right)^3} = 1124.864\]
\[\therefore\] We get the exact value of \[{\left( {10.4} \right)^3}\] as \[1124.864\].