Question

Express the following in decimal form: \[\dfrac{{23}}{{{2^4} \times {5^3}}}\].

Answer 1

Hint: Here, we need to express the given fraction in decimal form. First, we will use the rules of exponents to simplify the denominator. Then, we will convert the denominator to a product of 10s, since converting a fraction with denominator as product of 10s, to decimal form is easier. Then, we will rewrite the fraction in decimal form.

Formula Used: We will use the rule of exponent: If two or more numbers with same base and different exponents are multiplied, the product can be written as \[{a^b} \times {a^c} \times {a^d} = {a^{b + c + d}}\].

Complete step-by-step answer:
First, we will rewrite the denominator of the given fraction using the rule of exponents.
Rewriting \[\dfrac{{23}}{{{2^4} \times {5^3}}}\] as a fraction, we get
\[ \Rightarrow \dfrac{{23}}{{{2^4} \times {5^3}}} = \dfrac{{23}}{{{2^{1 + 1 + 1 + 1}} \times {5^{1 + 1 + 1}}}}\]
Therefore, rewriting the expression using the rule of exponents, \[{a^b} \times {a^c} \times {a^d} = {a^{b + c + d}}\], we get
\[ \Rightarrow \dfrac{{23}}{{{2^4} \times {5^3}}} = \dfrac{{23}}{{2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5}}\]
We know that the product of 2 and 5 is 10.
Pairing the terms in the denominator, we get
\[ \Rightarrow \dfrac{{23}}{{{2^4} \times {5^3}}} = \dfrac{{23}}{{2 \times \left( {2 \times 5} \right) \times \left( {2 \times 5} \right) \times \left( {2 \times 5} \right)}}\]
Multiplying the terms in the denominator, we get
\[ \Rightarrow \dfrac{{23}}{{{2^4} \times {5^3}}} = \dfrac{{23}}{{2 \times 10 \times 10 \times 10}}\]
It is easier to convert a fraction into a decimal if the denominator is a power of 10.
We will multiply both the numerator and denominator by 5.
Therefore, we get
\[\begin{array}{l} \Rightarrow \dfrac{{23}}{{{2^4} \times {5^3}}} = \dfrac{{23 \times 5}}{{2 \times 10 \times 10 \times 10 \times 5}}\\ \Rightarrow \dfrac{{23}}{{{2^4} \times {5^3}}} = \dfrac{{115}}{{10 \times 10 \times 10 \times 10}}\end{array}\]
Now, we can write the fraction in decimal form easily.
Multiplying the terms in the denominator, we get
\[ \Rightarrow \dfrac{{23}}{{{2^4} \times {5^3}}} = \dfrac{{115}}{{10000}}\]
Rewriting the fraction in decimal form, we get
\[\dfrac{{23}}{{{2^4} \times {5^3}}} = 0.0115\]
\[\therefore \] We have written the fraction \[\dfrac{{23}}{{{2^4} \times {5^3}}}\]as \[0.0115\] in decimal form.

Note: We have converted the denominator to a power of 10. This is because converting a fraction with a denominator to decimal is much easier. For example, in the fraction \[\dfrac{{23}}{{{2^4} \times {5^3}}} = \dfrac{{115}}{{10 \times 10 \times 10 \times 10}}\], 10 is multiplied 4 times. Therefore, we will add a decimal point before the fourth digit from the right in the numerator to get the decimal. We can rewrite 115 as 00115. Adding the decimal point before the fourth digit from the right, we get \[0.0115\]. Therefore, we can write \[\dfrac{{115}}{{10000}}\] as \[0.0115\] in decimal form.