Example Question - simplified expression
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Simplified Expression by Dividing Numerical Coefficients and Variables with Exponents
To simplify the given expression, divide the terms in the numerator by the terms in the denominator. The expression is: \[ \frac{7x^3y^{-5}}{21y^6} \] Divide the numerical coefficients: \[ \frac{7}{21} = \frac{1}{3} \] For the variables, use the laws of exponents to divide the terms with the same base: \[ x^{a} \div x^{b} = x^{a-b} \] \[ y^{a} \div y^{b} = y^{a-b} \] Applying the laws of exponents, you get: \[ x^3 \div x^0 = x^{3-0} = x^3 \] (Any term to the power of 0 is 1, which is why the x term in the denominator is considered \(x^0\)) \[ y^{-5} \div y^6 = y^{-5-6} = y^{-11} \] Now put it all together: \[ \frac{1}{3} x^3 y^{-11} \] The negative exponent indicates the reciprocal: \[ y^{-11} = \frac{1}{y^{11}} \] Thus, the simplified expression is: \[ \frac{x^3}{3y^{11}} \]
Simplified Expression with Square Roots and Cubes
The expression given in the image is: \[ \frac{2y\sqrt{x^3} + y^3 - 2x \times 2y}{2\sqrt{x^3} + y^3} \div \sqrt{x^3 + y^3} \] Let's simplify it step by step. First, we can simplify the numerator by combining like terms: \[2y\sqrt{x^3} + y^3 - 2x \times 2y = 2y\sqrt{x^3} + y^3 - 4xy\] Now, we recognize that \(\sqrt{x^3}\) is \(x^{3/2}\). This simplifies the expression to: \[ \frac{2yx^{3/2} + y^3 - 4xy}{2x^{3/2} + y^3} \div \sqrt{x^3 + y^3} \] Next, let's simplify the divisor. \(\sqrt{x^3 + y^3}\) is the root of the sum of cubes, and it is not factorable in the same simple way as the sum of squares. So, we leave it as is. Now we can rewrite the division as multiplication by the reciprocal: \[ \frac{2yx^{3/2} + y^3 - 4xy}{2x^{3/2} + y^3} \times \frac{1}{\sqrt{x^3 + y^3}} \] Now we multiply the fractions: \[ \frac{(2yx^{3/2} + y^3 - 4xy) \times 1}{(2x^{3/2} + y^3) \times \sqrt{x^3 + y^3}} \] Since there are no like terms and no obvious factorizations to simplify further, and we're not given values for \(x\) or \(y\), this seems to be as simplified as we can get without further instructions or context. Therefore, the simplified form of the given expression is: \[ \frac{2yx^{3/2} + y^3 - 4xy}{(2x^{3/2} + y^3) \sqrt{x^3 + y^3}} \]
Limit Calculation of Rational Function
The limit given in the image is \[ \lim_{{x \to 5}} \frac{x - 5}{x^2 - 25} \] We can start by factoring the denominator: \[ x^2 - 25 = (x - 5)(x + 5) \] The expression for the limit then becomes: \[ \lim_{{x \to 5}} \frac{x - 5}{(x - 5)(x + 5)} \] We can simplify the expression by canceling out the common factor of \(x - 5\) in the numerator and the denominator: \[ \lim_{{x \to 5}} \frac{1}{x + 5} \] Now we can directly substitute \(x = 5\) into the simplified expression, as there are no more discontinuities: \[ \frac{1}{5 + 5} = \frac{1}{10} \] So, the value of the limit is \(\frac{1}{10}\).
Solving an Algebraic Expression Using the Distributive Property
To solve the expression \( 6(1 - 5m) \), you need to apply the distributive property. This means that you'll multiply the number 6 by each term inside the parentheses. So, firstly, multiply 6 by 1, which gives you 6. Then, multiply 6 by \(-5m\), which gives you \(-30m\). Combine these two results to get the simplified expression: \[ 6 - 30m \]
Simplified Expression of 4c^2d * 3cd
The expression in the image is \(4c^2d \times 3cd\). To simplify the expression, you can multiply the numerical coefficients and add the exponents of the like variables. First, multiply the coefficients: \(4 \times 3 = 12\). Then, for the variables with exponents: - For c: \(c^2 \times c = c^{2+1} = c^3\). - For d: \(d \times d = d^{1+1} = d^2\). Now combine these together to get the simplified expression: \(12c^3d^2\).
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.
Simplified Expression with Factoring
To simplify the given expression, we will factor common terms and then reduce wherever possible. The expression is: (10 * 5^(n+4) + 125 * 5^(n+2)) / (3 * 5^(n+3) - 20 * 5^(n+1)) Let's factor out the common 5 raised to the smallest power in both the numerator and the denominator. For the numerator, the smallest power of 5 is n+2. We can then express each term with 5^(n+2) factored out: 5^(n+2) * (10 * 5^2 + 125) For the denominator, the smallest power of 5 is n+1. Factor out 5^(n+1): 5^(n+1) * (3 * 5^2 - 20) Now, we can substitute the powers of 5 with its actual value: Numerator becomes: 5^(n+2) * (10 * 25 + 125) 5^(n+2) * (250 + 125) 5^(n+2) * 375 Denominator becomes: 5^(n+1) * (3 * 25 - 20) 5^(n+1) * (75 - 20) 5^(n+1) * 55 Now, put the numerator and denominator back together: (5^(n+2) * 375) / (5^(n+1) * 55) Simplify by cancelling out the common factor of 5: 5^1 in the numerator remains after cancelling out with 5^(n+1) in the denominator and then also simplify the numbers: 5 * 375 / 55 Now, simplifying the numerical part, 375/55 can be reduced: 5 * 375 / 55 = 5 * 75 / 11 The final simplified expression is: (5 * 75) / 11 or 375 / 11
Algebraic Expression Simplification with Exponent Rules
This is an algebraic expression simplification problem involving exponent rules. We'll simplify each term by factoring out the common powers of 5 from the numerator and the denominator. Given expression: \( \frac{10 \times 5^{n+4} + 125 \times 5^{n+2}}{3 \times 5^{n+3} - 20 \times 5^{n+1}} \) Step 1: Factor out common powers of 5 from each term. Numerator: - Factor out \( 5^{n+2} \) from each term. \( 5^{n+2} (10 \times 5^2 + 125) = 5^{n+2} (10 \times 25 + 125) = 5^{n+2} (250 + 125) = 5^{n+2} \times 375 \) Denominator: - Factor out \( 5^{n+1} \) from each term. \( 5^{n+1} (3 \times 5^2 - 20) = 5^{n+1} (3 \times 25 - 20) = 5^{n+1} (75 - 20) = 5^{n+1} \times 55 \) Step 2: Simplify the expression with the factored out powers of 5. The simplified expression is: \( \frac{5^{n+2} \times 375}{5^{n+1} \times 55} \) Step 3: Cancel out the common power of 5 from the numerator and denominator. \( 5^{n+2} \) in the numerator and \( 5^{n+1} \) in the denominator have a common base, which means we can subtract the exponents. \( 5^{n+2 - (n+1)} = 5^1 = 5 \) So, the simplification yields: \( \frac{5 \times 375}{55} \) Step 4: Simplify the coefficients (375/55). Divide 375 by 55 to get: \( \frac{375}{55} = \frac{75}{11} \) So the final simplified expression is: \( \frac{5 \times 75}{11} \) Step 5: Multiply the 5 by 75. \( 5 \times 75 = 375 \) Therefore, the final simplified form of the expression is: \( \frac{375}{11} \) This fraction cannot be simplified further, as 375 and 11 do not have any common factors other than 1.
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