Example Question - simplified form
Here are examples of questions we've helped users solve.
Solving an Expression with Sum of Cubes
The image is rotated; I'll provide the steps to reach the solution as if the image were in the correct orientation. Given: a^2/b + b^2/a = ? To find the above expression, we combine the terms over a common denominator: Step 1: a^2/b + b^2/a = (a^3 + b^3) / (ab) Then, by applying the sum of cubes formula a^3 + b^3 = (a + b)(a^2 - ab + b^2), we simplify the expression: Step 2: (a^3 + b^3) / (ab) = [(a + b)(a^2 - ab + b^2)] / (ab) Finally, we factor out a from the first term and b from the second term in the numerator: Step 3: [(a + b)(a^2 - ab + b^2)] / (ab) = (a + b)(a - b + b) / b + (a + b)(a + a - b) / a This simplifies to: Step 4: (a + b)(a) / b + (a + b)(b) / a Now we split the terms to simplify further: Step 5: a(a + b) / b + b(a + b) / a Separate the terms: Step 6: a^2/b + ab/b + ab/a + b^2/a Further simplification: Step 7: a^2/b + a + a + b^2/a Combine like terms: Step 8: a^2/b + 2a + b^2/a And if required to write as a single fraction: Step 9: (a^3 + 2a^2b + b^3) / ab This represents the expression in terms of a single fraction, but the final simplified form is as mentioned in step 8: Final Answer: a^2/b + 2a + b^2/a
Simplifying a Fractional Expression
Bài toán yêu cầu chúng ta rút gọn biểu thức phân số sau: \[ \frac{5x + 10}{x^2 - 25} \] Để rút gọn, trước tiên chúng ta nhận ra rằng cả tử số và mẫu số đều có thể phân tích được: Tử số \(5x + 10\) có thể tách thành \(5(x + 2)\). Mẫu số \(x^2 - 25\) là hiệu của hai bình phương, có thể được phân tích thành \((x + 5)(x - 5)\) sử dụng công thức \(a^2 - b^2 = (a + b)(a - b)\). Khi đó, biểu thức có thể rút gọn như sau: \[ \frac{5(x + 2)}{(x + 5)(x - 5)} \] Xem xét mối quan hệ giữa tử số và mẫu số, chúng ta thấy không có yếu tố chung nào để có thể rút gọn thêm. Do đó, đây chính là dạng rút gọn nhất của biểu thức phân số trên với điều kiện \(x \neq \pm5\) (điều kiện này đảm bảo mẫu số không bằng 0).
Simplifying Square Roots with Negative Sign
The expression shown in the image is "-√12". To solve this, we will simplify the square root of 12 first and then apply the negative sign. √12 can be broken down into √(4 * 3) = √4 * √3. Since we know that √4 is 2, we have √12 = 2√3. Applying the negative sign to this result, we get: -√12 = -2√3 So, the simplified form of "-√12" is "-2√3".
Simplifying Mathematical Expressions
The image shows a mathematical expression that needs to be simplified: (4X^5)^3 + (2X^3)^4 To simplify this expression, you have to raise each term inside the parentheses to the power outside the parentheses. This is done by raising both the coefficient and the variable to the power. Remember that (a^m)^n = a^(mn) when you raise a power to a power. (4X^5)^3: Raising 4 to the 3rd power gives 64. Raising X^5 to the 3rd power gives X^(5*3), which is X^15. (2X^3)^4: Raising 2 to the 4th power gives 16. Raising X^3 to the 4th power gives X^(3*4), which is X^12. So, the expression simplifies as follows: (4X^5)^3 + (2X^3)^4 = 64X^15 + 16X^12 That is the simplified form of the expression provided in the image.
Solving Expressions Using Distributive Property
To solve the expression given in the image, you'll need to apply the distributive property, which states that \( a(b + c) = ab + ac \). So for the expression \( 4(8n + 2) \), use the distributive property as follows: \( 4 \times 8n + 4 \times 2 \) Now multiply the numbers: \( 32n + 8 \) The simplified form of the expression is \( 32n + 8 \).
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 Expressions with Exponents
Sure, to simplify the expression provided in the image, we need to follow the rules of algebra and exponents. The expression in the image is: (10 × 5^n^4 + 125 × 5^n^2) / (3 × 5^n^3 + 20 × 5^n^1) Let's simplify the numerator and the denominator separately. First, we can factor out the greatest common factor in the numerator, which is 5^n^2: = 5^n^2 (10 × 5^n^2 + 125) / (3 × 5^n^3 + 20 × 5^n^1) And then we can do the same for the denominator, where the greatest common factor is 5^n^1: = 5^n^2 (10 × 5^n^2 + 125) / 5^n^1 (3 × 5^n^2 + 20) Now, let's simplify the terms inside the parentheses: Numerator: 10 × 5^n^2 + 125 can be simplified to 5^n^2 + 125. Since no further simplification can be done inside the parentheses (assuming the "+ 125" is not supposed to have a power of 5 associated with it), let's keep it as is. Denominator: 3 × 5^n^2 + 20 simplifies to 3 × 5^n^2 + 20. Again, assuming "20" does not have a power of 5 associated with it, there's no further simplification inside the parentheses. So now we have: = 5^n^2 (5^n^2 + 125) / 5^n^1 (3 × 5^n^2 + 20) Now, let's simplify across the numerator and denominator by canceling common factors where possible. We can cancel one 5^n^1 from both the numerator and the denominator: = 5^n (5^n^2 + 125) / (3 × 5^n^2 + 20) We cannot simplify further without knowing the value of n. If n is such that n^2, n^1, and the simple numerical terms 125 and 20 can be simplified further, we'd do that. Otherwise, this is the simplified form of the given expression. Please note that the superscripts in the original expression might be ambiguous due to the resolution and angle of the photo. If the powers are not correctly interpreted, please provide more information or adjust the expression accordingly.
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.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com