Question

How do you simplify$\left( 9\times {{10}^{7}} \right)\left( 7\times {{10}^{9}} \right)$ ?

Answer 1

Hint: The above given equation $\left( 9\times {{10}^{7}} \right)\left( 7\times {{10}^{9}} \right)$ is simplify in the $\left( a\times b \right)$ form. And it is in exponent and power form. Exponents and powers are ways used to represent very large numbers or very small numbers in a simplified manner. Exponent tells us how many times a number should be multiplied by itself to get the desired result. For example, thus any number `a` raised to power `n` can be expressed as $\Rightarrow {{a}^{n}}=a\times a\times a\times .......a\left( n-times \right)$.
There are many rules of exponents which are like
Multiplication rule: ${{a}^{x}}\times {{a}^{y}}={{a}^{x+y}}$,
Division rule: ${{a}^{x}}\div {{a}^{y}}={{a}^{x-y}}$,
Power of power rule: ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}$,
Power of a product rule: ${{\left( ab \right)}^{x}}={{a}^{x}}{{b}^{x}}$,
Power of a fraction rule: ${{\left( \dfrac{a}{b} \right)}^{x}}=\dfrac{{{a}^{x}}}{{{b}^{x}}}$,
Zero Exponent: ${{a}^{0}}=1$, Negative exponent:${{a}^{-x}}=\dfrac{1}{{{a}^{x}}}$,
Fractional exponent: ${{a}^{\dfrac{x}{y}}}=\sqrt[y]{{{a}^{x}}}$.
Here we will only use multiplication exponent rule which is ${{a}^{x}}\times {{a}^{y}}={{a}^{x+y}}$.

Complete step by step solution:
The given equation is:
$\Rightarrow \left( 9\times {{10}^{7}} \right)\left( 7\times {{10}^{9}} \right)$
To simplify this we will do some steps. We will simply multiply $9\times 7$ and we have to add the powers of the $10$ which are $7+9$
$\Rightarrow \left( 9\times 7 \right)\left( {{10}^{7+9}} \right)$
We know $9\times 7=63$ and ${{10}^{7+9}}={{10}^{16}}$, then we get
$\Rightarrow 63\times {{10}^{16}}$
Hence by using the simple rule of exponent, we get the simplified form of the above given equation $\left( 9\times {{10}^{7}} \right)\left( 7\times {{10}^{9}} \right)$ is $63\times {{10}^{16}}$.

Note: To solve these types of expressions we should know the basic rules of powers. Here only one rule is used. We have expression in the multiplication form and if we have any two same numbers with different powers then we will simply add the powers of the same numbers ${{a}^{x}}\times {{a}^{y}}={{a}^{x+y}}$. So whenever we solve these types of questions we must know these formulas otherwise our solution can be wrong.