Example Question - simplify algebraic expression
Here are examples of questions we've helped users solve.
Simplification of an Algebraic Fraction with Negative Exponents
\[ \left( \frac{4a^6b^5c^{-2}}{(2a^{-4}b^6c)^3} \right)^2 = \left( \frac{4a^6b^5c^{-2}}{8a^{-12}b^{18}c^3} \right)^2 = \left( \frac{4 \cdot a^{6-(-12)} \cdot b^{5-18} \cdot c^{-2-3}}{8} \right)^2 = \left( \frac{4 \cdot a^{18} \cdot b^{-13} \cdot c^{-5}}{8} \right)^2 = \left( \frac{a^{18} \cdot b^{-13} \cdot c^{-5}}{2} \right)^2 = \frac{a^{36} \cdot b^{-26} \cdot c^{-10}}{4} \]
Simplifying an Algebraic Expression
To simplify the expression \( 3p(p - q) - (2p - q)^2 \), follow these steps: Step 1: Distribute the \(3p\) in the first term: \[ 3p^2 - 3pq \] Step 2: Expand the squared term \((2p - q)^2\): \[ (2p - q)(2p - q) = 4p^2 - 4pq + q^2 \] Step 3: Subtract the expanded squared term from the first term: \[ (3p^2 - 3pq) - (4p^2 - 4pq + q^2) \] Step 4: Distribute the negative sign across the terms in the parentheses: \[ 3p^2 - 3pq - 4p^2 + 4pq - q^2 \] Step 5: Combine like terms: \[ (3p^2 - 4p^2) + (-3pq + 4pq) - q^2 \] \[ -p^2 + pq - q^2 \] So, \( 3p(p - q) - (2p - q)^2 = -p^2 + pq - q^2 \).
Simplifying an Algebraic Expression
To simplify the given algebraic expression: \[\frac{\frac{2}{x} - 5}{6 + \frac{3}{x}}\] First, find a common denominator for the fractions in the numerator and the denominator. The common denominator for the fractions with \(x\) in the numerator is \(x\), and similarly for the denominator. Rewrite the expression with common denominators as follows: \[\frac{\left(\frac{2}{x} - \frac{5x}{x}\right)}{\left(6\frac{x}{x} + \frac{3}{x}\right)}\] This simplifies to: \[\frac{\frac{2 - 5x}{x}}{\frac{6x + 3}{x}}\] Now, since the denominators are the same, the fractions can be combined: \[\frac{2 - 5x}{6x + 3}\] This is the simplified form of the given expression. It cannot be further simplified as there are no common factors between the numerator and the denominator that can be cancelled out.
Simplifying Algebraic Expression with Like Terms
The given expression is: \( 2a^2b - 8ab^2 + b^4 + 3ab^2 \) To simplify the expression, combine like terms. Like terms are terms that have the exact same variables raised to the exact same powers. The first term \(2a^2b\) does not have any other like terms, so we can leave it as it is. The second term \(-8ab^2\) and the fourth term \(3ab^2\) are like terms because they both have \(ab^2\). Combine them: \(-8ab^2 + 3ab^2 = -5ab^2\) The third term \(b^4\) does not have any like terms, so we also leave that term as it is. Putting them all together, our simplified expression is: \( 2a^2b - 5ab^2 + b^4 \) That’s the final simplified form of the given expression.
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 Algebraic Expression by Factoring
Certainly! To simplify the given expression, we need to factor out common terms and potentially simplify both the numerator and the denominator. The expression is: (10x^(5n+4) + 125x^(5n+2)) / (3x^(5n+3) - 20x^(5n+1)) Now, let's simplify both parts: The numerator can be factored by x^(5n+2), which is the smallest power of x present in both terms: x^(5n+2)(10x^2 + 125) Similarly, the denominator can be factored by x^(5n+1): x^(5n+1)(3x^2 - 20) After factoring out, we get: x^(5n+2)(10x^2 + 125) / x^(5n+1)(3x^2 - 20) Now, we can reduce the expression by eliminating common factors. Notice that x^(5n+2) in the numerator and x^(5n+1) in the denominator can be simplified: x^(5n+2) / x^(5n+1) = x ((5n+2) - (5n+1)) = x^(1) Now we have: x(10x^2 + 125) / (3x^2 - 20) This cannot be further simplified without knowing the specific values of x or unless the quadratic expressions factor into terms that cancel each other out (which is not the case here). Thus the simplified expression is: x(10x^2 + 125) / (3x^2 - 20)
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