Example Question - simplify fraction exponents
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Simplifying a Fraction with Exponents
The expression in the image is a fraction where the numerator is \(10 \times 5^{n+4} + 125 \times 5^{n+2}\) and the denominator is \(3 \times 5^{n+3} + 20 \times 5^{n+1}\). To solve it, you can use the properties of exponents to simplify the expression. 1. Factor out the common \(5^n\) term in both numerator and denominator: Numerator: \(5^n(10 \times 5^4 + 125 \times 5^2)\) Denominator: \(5^n(3 \times 5^3 + 20 \times 5^1)\) 2. Now, cancel out the \(5^n\) term from both the numerator and denominator: Numerator simplifies to: \(10 \times 5^4 + 125 \times 5^2\) Denominator simplifies to: \(3 \times 5^3 + 20 \times 5^1\) 3. Simplify the coefficients and the remaining powers of 5: Numerator: \(10 \times 625 + 125 \times 25 = 6250 + 3125 = 9375\) Denominator: \(3 \times 125 + 20 \times 5 = 375 + 100 = 475\) 4. Simplify the fraction if possible. In this case, 9375 and 475 don't share any common factors other than 1, so the fraction is already in its simplest form: Final simplified expression: \(\frac{9375}{475}\) The final answer is a simplified fraction or you could also convert it to a decimal or mixed number if you divide 9375 by 475.
Simplifying a Fraction with Exponents
The expression given is: \[ \frac{10 \cdot 5^{n+4} + 125 \cdot 5^{n+2}}{3 \cdot 5^{n+3} - 20 \cdot 5^{n+1}} \] To simplify, we'll factor out the common base of \( 5^n \) from both the numerator and the denominator: \[ = \frac{5^n (10 \cdot 5^4 + 125 \cdot 5^2)}{5^n (3 \cdot 5^3 - 20 \cdot 5)} \] \[ = \frac{5^n (10 \cdot 625 + 125 \cdot 25)}{5^n (3 \cdot 125 - 20 \cdot 5)} \] \[ = \frac{5^n (6250 + 3125)}{5^n (375 - 100)} \] \[ = \frac{5^n \cdot 9375}{5^n \cdot 275} \] Since \( 5^n \) is a common factor in both the numerator and the denominator, we can cancel it out: \[ = \frac{9375}{275} \] Now we'll divide both the numerator and the denominator by 25 to simplify further: \[ = \frac{375}{11} \] The simplified form of the expression is \( \frac{375}{11} \), which cannot be simplified any further to an integer or a reduced fraction.
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