Question

The distance between the earth and the sun is $150$ million kilometers. This distance can be written in scientific notation as $1.5\times {10}^x$ kilometers. The value of $x$ is .

Answer 1

Given:

• The distance between the earth and the sun is $150$ million kilometers.
• The scientific notation of $150$ million kilometers is $1.5\times {10}^x$ kilometers.

Expressing $150$ million kilometers in scientific notation

Step 1:
Write the given distance in exponent form.
$150$ million kilometers can be written as $150.0\times {10}^6$ in exponent form.

Step 2:
Move the decimal point $2$ places to the left so that the decimal point positioned to the right of the leftmost non-zero digit $1.$


Step 3:
Since the decimal point is moved $2$ places to the left, we use here a positive integer $2$ as an exponent to the base $10$.

$\therefore 150.0\times {10}^6=1.5\times {10}^2\times {10}^6$

Applying exponent rule: $a^m\times a^n=a^{(m+n)}$

$=1.5\times {10}^{(2+6)}$

$=1.5\times {10}^8$

Therefore, $150$ million kilometers can be written as $1.5\times {10}^8$ kilometers in scientific notation.

Now, according to the question:
$1.5\times {10}^x=1.5\times {10}^8$

In the above equation, since the coefficients and bases are the same, for equality, the exponents must also be the same.
$⟹x=8$

Therefore, the value of $x$ is $8$.