Question

Given that, \[512={{8}^{3}}\] and \[3.375={{1.5}^{3}}\], find the value of \[\sqrt[3]{512}\times \sqrt[3]{3.375}\]
A.12
B.9.5
C.8
D.1.5

Answer 1

Hint: A cube root of number x is a number y such that \[{{y}^{3}}=x\]all non-zero numbers, have exactly one cube root. Here, we have given the two cube root values and then we have to find the \[\sqrt[3]{512}\times \sqrt[3]{3.375}\] by substituting the given values.

Complete step-by-step solution:
Generally, in mathematics the term cube root of a number is a value which when multiplied by it thrice or simply which when multiplied by itself three times produces the original value.
In mathematical notation, let there be a number a then the cube root of a will be \[\sqrt[3]{a}\].
For the given question,
we have given that \[512={{8}^{3}}\] and \[3.375={{1.5}^{3}}\]
Now we have to find out the value of \[\sqrt[3]{512}\times \sqrt[3]{3.375}\]
To find out the above value, we need to substitute the values of \[512={{8}^{3}}\] and \[3.375={{1.5}^{3}}\]
Then we get as,
\[\Rightarrow \sqrt[3]{{{8}^{3}}}\times \sqrt[3]{{{\left( 1.5 \right)}^{3}}}\]
On cancelling the cube root and root of a number, we will obtain as
\[\Rightarrow 8\times 1.5\]
Which is equal to 12.
\[\therefore \sqrt[3]{512}\times \sqrt[3]{3.375}=12\]
Hence, the correct option is (A).
We can solve this problem by using the exponent formula.
We have the formula of cube roots of the product of two number as
\[\Rightarrow \sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}\]
\[\Rightarrow \sqrt[3]{{{8}^{3}}}\times \sqrt[3]{{{\left( 1.5 \right)}^{3}}}\]
\[\Rightarrow \sqrt[3]{{{\left( 8\times 1.5 \right)}^{3}}}\]
\[\Rightarrow \left( 8\times 1.5 \right)=12\]
Hence, the correct option is (A).

Note: Knowing the concepts is a very crucial thing in these kinds of questions. And also, you should be very good in multiples. As, ultimately your answer depends on the multiplication of the numbers. Also, know the difference between the cubes and cube roots and apply accordingly. Cube root can be defined as the number which produces a given number when cubed.