Question

How do you simplify $\left( -4{{b}^{3}} \right)\left( 2{{b}^{5}} \right)$

Answer 1

Hint: Now to simplify the expression we will first multiply the constant terms in the expression. Now we will simplify the expression by using the multiplication law of indices which states ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$ . Hence we will simplify the given expression.

Complete step by step solution:
Let us understand the concept of indices.
Indices or index is nothing but a number raised to a terms.
The number raised is called power of the term.
Now the term can be variable or a number.
For example let us consider ${{a}^{3}}$ here 3 is raised to a and hence power is 3.
Now let us understand what the power means. The power represents the number of times the term is multiplied by itself.
Hence if we have ${{2}^{4}}$ then we can write it as $2\times 2\times 2\times 2$ .
Now let us understand the laws of indices.
The multiplication law of indices states that ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$
The division law of indices states $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
The exponent law of indices states ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ .
Now consider the given expression $\left( -4{{b}^{3}} \right)\left( 2{{b}^{5}} \right)$
First we will multiply the constants in the expression. Hence we get, $-8{{b}^{3}}{{b}^{5}}$
Now we know according to multiplication law of indices we get ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$
Hence using this in the above expression we get, $-8{{b}^{5+3}}$ .
On simplification we get the expression as $-8{{b}^{8}}$
Hence the given expression can be written as $-8{{b}^{8}}$ .

Note:
Now note that the power of index can be negative number or zero also. For any number a we have the value of ${{a}^{0}}$ is 1. For negative integers we can write the term in denominator and raise it to the positive power of the same magnitude. Hence we have ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ .