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Simplify $\dfrac{5^x \times 7-5^x}{5^{x+2}-5^{x+1}}$

It is a function in fractional form, which is formed by the quotient of two exponential expressions $5^{\displaystyle x} \times 7-5^{\displaystyle x}$ and $5^{{\displaystyle x}+2}-5^{{\displaystyle x}+1}$, where $x$ represents a variable. In this problem, it is given that the value of the given function has to be calculated by simplification.

Split each exponential term

The following given function can be simplified by simplifying the exponential expressions in the both numerator and denominator of the fractional function.

$\dfrac{5^{\displaystyle x} \times 7-5^{\displaystyle x}}{5^{{\displaystyle x}+2}-5^{{\displaystyle x}+1}}$

In numerator, the exponential expression is formed purely by exponential terms but the exponential expression in denominator is formed by the terms which contain expressions as exponents. Mathematically, the numerator can be simplified easily but the expression in the denominator can only be simplified by splitting each term which contains an expression as an exponent. It can be done by using the product rule of exponents with same base.

$= \,\,\,$ $\dfrac{5^{\displaystyle x} \times 7-5^{\displaystyle x}}{5^{\displaystyle x} \times 5^2-5^{\displaystyle x} \times 5^1}$

$= \,\,\,$ $\dfrac{5^{\displaystyle x} \times 7-5^{\displaystyle x} \times 1}{5^{\displaystyle x} \times 5^2-5^{\displaystyle x} \times 5^1}$

Take same factors out from expression

$5^{\displaystyle x}$ is a factor in each term of exponential expression in both numerator and denominator. According to the distributive property of multiplication over subtraction, it can be taken out as a common factor from the exponential terms in each expression for simplifying the whole function.

$= \,\,\,$ $\dfrac{5^{\displaystyle x} \times (7-1)}{5^{\displaystyle x} \times (5^2-5^1)}$

Simply each exponential expression

In denominator, two exponents with same base are involved in subtraction but it is not possible to subtract them directly. So, write the value of each exponential term.

$= \,\,\,$ $\dfrac{5^{\displaystyle x} \times (7-1)}{5^{\displaystyle x} \times (25-5)}$

In both numerator and denominator, there are two natural numbers involved in subtraction in every exponential expression. So, subtract them as per subtraction of natural numbers.

$= \,\,\,$ $\dfrac{5^{\displaystyle x} \times 6}{5^{\displaystyle x} \times 20}$

Use the division of fractions for eliminating the same factors from both numerator and denominator of the fraction.

$= \,\,\,$ $\require{cancel} \dfrac{\cancel{5^{\displaystyle x}} \times 6}{\cancel{5^{\displaystyle x}} \times 20}$

$= \,\,\,$ $\dfrac{1 \times 6}{1 \times 20}$

$= \,\,\,$ $\dfrac{6}{20}$

$= \,\,\,$ $\require{cancel} \dfrac{\cancel{6}}{\cancel{20}}$

$= \,\,\,$ $\dfrac{3}{10}$

Therefore, it is simplified that the value of the ratio of exponential expressions $5^{\displaystyle x} \times 7-5^{\displaystyle x}$ to $5^{{\displaystyle x}+2}-5^{{\displaystyle x}+1}$ is equal to $\dfrac{3}{10}$.

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