Example Question - factor out common terms
Here are examples of questions we've helped users solve.
Simplifying a Rational Expression using Exponent Properties
To simplify the given expression, we will apply the properties of exponents. The expression is: \[ \frac{10x^{n+4} + 125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \] First, factor out the common factor of \(x\) from the numerator and denominator: Numerator: Take \( x^{n+2} \) as the common factor. \[ x^{n+2} (10x^2 + 125) \] Denominator: Take \( x^{n+1} \) as the common factor. \[ x^{n+1} (3x^2 - 20) \] Now the expression becomes: \[ \frac{x^{n+2} (10x^2 + 125)}{x^{n+1} (3x^2 - 20)} \] Now we will cancel the common \(x\) terms, remembering that \(x^{n+2} / x^{n+1} = x\): \[ \frac{x (10x^2 + 125)}{3x^2 - 20} \] After canceling, we are then left with the simplified expression that we cannot further simplify: \[ \frac{x (10x^2 + 125)}{3x^2 - 20} \] This is the simplified version of the original expression, assuming that \(x \neq 0\) and \(x^2 \neq \frac{20}{3}\) so that the denominator is not zero.
Simplifying an Algebraic Expression with Exponents and Division
To simplify the given expression, we follow the rules of exponents and division. The expression is: \[ \frac{10x^{5n+4} + 125x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} \] Let's break it down step by step: First, factor out the common factor of x in both numerator and denominator. In the numerator, the smallest power of x is \(x^{5n+2}\), and in the denominator, the smallest power is \(x^{5n+1}\). We factor these out from both parts: Numerator factor: \(x^{5n+2}\) (leaving \(10x^2 + 125\)) Denominator factor: \(x^{5n+1}\) (leaving \(3x^2 - 20\)) After factoring out, we have: \[ \frac{x^{5n+2}(10x^2 + 125)}{x^{5n+1}(3x^2 - 20)} \] Now we can simplify \(x^{5n+2}\) and \(x^{5n+1}\). Subtracting the exponents (using the properties of exponents), we get \(x^{5n+2 - (5n+1)} = x^{(5n+2) - 5n - 1} = x^1 = x\). The simplified form of the expression is: \[ \frac{x(10x^2 + 125)}{3x^2 - 20} \] We cannot simplify any further without knowing the specific values for x. This is the simplified expression in terms of x and n.
Simplifying Expression with Exponents
To simplify the expression given in the image, we need to follow the laws of exponents for division and addition. The expression is: \[ \frac{10x^{n+4} + 4.125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \] First, factor out the common terms involving \(x\) from the numerator and the denominator. 1. In the numerator, factor out the smallest power of \(x\) which is \(x^{n+2}\): \[ x^{n+2}(10x^{2} + 4.125) \] 2. In the denominator, factor out the smallest power of \(x\) which is \(x^{n+1}\): \[ x^{n+1}(3x^{2} - 20) \] Now we rewrite the expression: \[ \frac{x^{n+2}(10x^{2} + 4.125)}{x^{n+1}(3x^{2} - 20)} \] Next, cancel out the common \(x^{n+1}\) term: \[ \frac{x(10x^{2} + 4.125)}{3x^{2} - 20} \] Since \(x\) does not cancel anything further in the denominator, we leave the expression as is. The final simplified expression is: \[ \frac{x(10x^{2} + 4.125)}{3x^{2} - 20} \] Depending on the context or further instructions, there might be additional simplification possible, like factoring the quadratic expressions if they are factorable. However, without additional context, this is the simplified form based on the given expression.
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