Question

How do you find the product of \[{\left( {4u + 9v} \right)^2}\]?

Answer 1

Hint: The algebraic expression should be any one of the forms such as addition, subtraction, multiplication and division and to find the product of the equation, as it is in the form \[{\left( {a + b} \right)^2}\]we know its expansion as \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\], hence apply the formula to get the product.

Complete step-by-step solution:
Let us write the given equation
\[\Rightarrow {\left( {4u + 9v} \right)^2}\]
As the given equation is of the form \[{\left( {a + b} \right)^2}\], hence let us equate and find the product by expanding the terms of the given equation.
We know that
\[\Rightarrow {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
Substituting the value of a and b as
\[\Rightarrow {\left( {4u} \right)^2} + 2\left( {4u} \right)\left( {9v} \right) + {\left( {9v} \right)^2}\]
Hence, evaluating the terms we get
\[\Rightarrow 16{u^2} + 72uv + 81{v^2}\]
Therefore, the product of \[{\left( {4u + 9v} \right)^2}\] is
\[\Rightarrow 16{u^2} + 72uv + 81{v^2}\].

Hence the answer is \[16{u^2} + 72uv + 81{v^2}\].

Additional information: There are four basic properties of multiplication
Commutative property: When two numbers are multiplied together, the product is the same regardless of the order in which they are written.
Associative property: When three or more numbers are multiplied together, the product is the same regardless of which two are multiplied first.
Multiplicative identity property: This says that any number multiplied by 1 result in the same number you had before. 1 is called a multiplicative identity.
Distributive Property: The sum of two numbers times a third number is equal to sum of each times the third number.
Closure Property: According to this property, if two integers a and b are multiplied then their resultant $a\times b$ is also an integer.
Multiplication by zero: On multiplying any integer by zero the result is always zero.
Multiplicative Identity of Integers: On multiplying any integer by 1 the result obtained is the integer itself.

Note: Multiplication is a method of finding the product of two or more values. In arithmetic, multiplication of two numbers represents the repeated addition of one number with respect to another. Integers are the whole numbers but it does not include fractions. The integer can be either positive integer or negative integer.