Example Question - simplify

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Simplifying a Mathematical Expression

<p>Start with the expression:</p> <p>\(\frac{15^{16}}{15^{4} \times (15^{2})^{3}}\)</p> <p>First, simplify \((15^{2})^{3}\):</p> <p>\((15^{2})^{3} = 15^{6}\)</p> <p>Now substitute back into the expression:</p> <p>\(\frac{15^{16}}{15^{4} \times 15^{6}}\)</p> <p>Combine the terms in the denominator:</p> <p>So, the denominator becomes \(15^{4 + 6} = 15^{10}\)</p> <p>Now the expression is:</p> <p>\(\frac{15^{16}}{15^{10}}\)</p> <p>Using the quotient rule of exponents:</p> <p>Subtract the exponents: \(15^{16 - 10} = 15^{6}\)</p> <p>Thus, the simplified expression is:</p> <p>\(15^{6}\)</p>

Simplifying a Fraction with Exponents

<p>Given the expression:</p> <p>\[\frac{10^{-8}}{7^5 \times 10^3 \times 7^{-7}}\]</p> <p>We can rewrite the denominator:</p> <p>\(7^5 \times 10^3 \times 7^{-7} = 10^3 \times 7^{5 - 7} = 10^3 \times 7^{-2}\)</p> <p>Next, we place this back into the expression:</p> <p>\[\frac{10^{-8}}{10^3 \times 7^{-2}} = \frac{10^{-8}}{10^3} \times 7^2\]</p> <p>Simplifying \(\frac{10^{-8}}{10^3}\):</p> <p>\(10^{-8 - 3} = 10^{-11}\)</p> <p>The expression becomes:</p> <p>\(10^{-11} \times 7^2\)</p> <p>Thus, the final simplified expression is:</p> <p>\[7^2 \times 10^{-11}\]</p> <p>Which can be written as:</p> <p>\(49 \times 10^{-11}\)</p>

Simplifying Exponential Expressions

<p>To simplify \(10^{-8}\), we can express it as:</p> <p>\(10^{-8} = \frac{1}{10^{8}}\)</p>

Simple Fraction Multiplication

<p>\( \frac{4}{7} \times \frac{5}{2} = \frac{4 \times 5}{7 \times 2} \)</p> <p>\( = \frac{20}{14} \)</p> <p>\( = \frac{10}{7} \) (بعد اختصار الكسر بالقسمة على 2)</p>

Fraction Operations and Order of Operations

<p>Первое действие:</p> <p>\[ \frac{12}{19} - \left( \frac{7}{12} - \frac{4 \cdot 13}{21} \right) \]</p> <p>\[ \frac{12}{19} - \left( \frac{7}{12} - \frac{52}{21} \right) \]</p> <p>Найти общий знаменатель для \(\frac{7}{12}\) и \(\frac{52}{21}\):</p> <p>\[ 12 \cdot 21 = 252 \]</p> <p>\[ \frac{12}{19} - \left( \frac{7 \cdot 21}{252} - \frac{52 \cdot 12}{252} \right) \]</p> <p>\[ \frac{12}{19} - \left( \frac{147}{252} - \frac{624}{252} \right) \]</p> <p>\[ \frac{12}{19} - \left( -\frac{477}{252} \right) \]</p> <p>\[ \frac{12}{19} + \frac{477}{252} \]</p> <p>Теперь найдем общий знаменатель для \(\frac{12}{19}\) и \(\frac{477}{252}\):</p> <p>\[ 19 \cdot 252 = 4788 \]</p> <p>\[ \frac{12 \cdot 252}{4788} + \frac{477 \cdot 19}{4788} \]</p> <p>\[ \frac{3024}{4788} + \frac{9063}{4788} \]</p> <p>\[ \frac{12087}{4788} \]</p> <p>Упрощаем дробь:</p> <p>\[ \frac{12087 \div 3}{4788 \div 3} \]</p> <p>\[ \frac{4029}{1596} \]</p> <p>Упрощаем дробь:</p> <p>\[ \frac{4029 \div 3}{1596 \div 3} \]</p> <p>\[ \frac{1343}{532} \]</p> <p>Далее применяем деление с остатком:</p> <p>\[ 1343 = 532 \cdot 2 + 279 \]</p> <p>\[ \frac{1343}{532} = 2 \frac{279}{532} \]</p> <p>Второе действие:</p> <p>\[ \left( 3 \frac{2}{7} - 25,8 \right) \cdot \frac{7}{23} \]</p> <p>Преобразуем смешанное число в неправильную дробь:</p> <p>\[ 3 \frac{2}{7} = \frac{3 \cdot 7 + 2}{7} = \frac{21 + 2}{7} = \frac{23}{7} \]</p> <p>\[ \left( \frac{23}{7} - 25,8 \right) \cdot \frac{7}{23} \]</p> <p>Переведем 25,8 в дробь:</p> <p>\[ 25,8 = \frac{258}{10} = \frac{129}{5} \]</p> <p>Теперь найдем общий знаменатель для \(\frac{23}{7}\) и \(\frac{129}{5}\):</p> <p>\[ 7 \cdot 5 = 35 \]</p> <p>\[ \left( \frac{23 \cdot 5}{35} - \frac{129 \cdot 7}{35} \right) \cdot \frac{7}{23} \]</p> <p>\[ \left( \frac{115}{35} - \frac{903}{35} \right) \cdot \frac{7}{23} \]</p> <p>\[ \left( -\frac{788}{35} \right) \cdot \frac{7}{23} \]</p> <p>\[ -\frac{5516}{35 \cdot 23} \]</p> <p>\[ -\frac{5516}{805} \]</p> <p>Применяем деление с остатком:</p> <p>\[ 5516 = 805 \cdot 6 + 746 \]</p> <p>\[ -\frac{5516}{805} = -6 \frac{746}{805} \]</p> <p>Ответы на задания:</p> <p>1) \( 2 \frac{279}{532} \)</p> <p>2) \( -6 \frac{746}{805} \)</p>

Solving a Simple Algebraic Fraction

<p>Primero simplificamos la expresión algebraica en el numerador:</p> <p>\[ (y + 7) - 6(-1) = y + 7 + 6 \]</p> <p>Ahora combinamos los términos semejantes:</p> <p>\[ y + 13 \]</p> <p>Luego, escribimos la expresión simplificada sobre el denominador:</p> <p>\[ \frac{y + 13}{3} \]</p> <p>Esta es la expresión simplificada y no se puede simplificar más ya que \( y \) es una variable y no sabemos su valor.</p>

Subtracting Fractions with Different Denominators

<p>Para restar fracciones con diferentes denominadores, primero encontramos un denominador común y luego restamos los numeradores. En este caso, el denominador común más pequeño para \( 36 \) y \( 81 \) es \( 36 \times 9 = 324 \).</p> <p>\( \frac{48}{36} = \frac{48 \times 9}{36 \times 9} = \frac{432}{324} \)</p> <p>\( \frac{18}{81} = \frac{18 \times 4}{81 \times 4} = \frac{72}{324} \)</p> <p>Ahora que tienen el mismo denominador, podemos restar los numeradores:</p> <p>\( \frac{432}{324} - \frac{72}{324} = \frac{432 - 72}{324} = \frac{360}{324} \)</p> <p>Finalmente, simplificamos la fracción dividiendo tanto el numerador como el denominador por su máximo común divisor, que es \( 36 \):</p> <p>\( \frac{360}{324} = \frac{360 \div 36}{324 \div 36} = \frac{10}{9} \)</p>

Simplifying a Rational Expression Involving Polynomials

<p> \( y = \frac{x}{(2x-1) \cdot (3) \cdot (x-1)} \) </p> <p> To simplify the expression, factor out and reduce common terms if any. </p> <p> No common terms to factor out or reduce. </p> <p> The expression is already in its simplest form. So, the simplified expression is: </p> <p> \( y = \frac{x}{6x^2 - 3x - 2x + 1} \) </p> <p> \( y = \frac{x}{6x^2 - 5x + 1} \) </p>

Simplify Radical and Fractional Expression

<p>\( \sqrt{8} + \frac{4}{\sqrt{2}} \)</p> <p>\( = \sqrt{4 \cdot 2} + \frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \)</p> <p>\( = 2\sqrt{2} + \frac{4\sqrt{2}}{2} \)</p> <p>\( = 2\sqrt{2} + 2\sqrt{2} \)</p> <p>\( = 4\sqrt{2} \)</p>

Simplified Algebraic Expression with Fractions and Exponents

The image contains an algebraic expression within a square root and two fractions added together underneath. Both the numerator and denominator of the fractions contain terms with x raised to powers and multiplied by y, and y multiplied by x raised to powers respectively. To solve or simplify the expression, let's consider each fraction separately and observe if there's a common pattern. Starting with the first fraction: `x^(2x-3y) / x^(3y-2x)`. We can simplify this using the laws of exponents. When we divide two exponents with the same base, we subtract the exponent in the denominator from the exponent in the numerator: `x^(a)/x^(b) = x^(a-b)`. Applying this law to the first fraction, we have: `x^(2x-3y - (3y-2x)) = x^((2x-3y) - 3y + 2x) = x^(4x-6y)`. For the second fraction `y^(2x-3y) / y^(3y-2x)`, the same rules apply: `y^(2x-3y - (3y-2x)) = y^((2x-3y) - 3y + 2x) = y^(4x-6y)`. Now adding the two simplified terms together: `x^(4x-6y) + y^(4x-6y)`. Under the square root, the expression cannot be simplified further algebraically without knowing the values of x and y. If x and y are equal, then we could factor out `(4x-6y)` as a common term, but since the expression doesn't provide that information, this is the simplest form. Therefore, the simplified form of the expression under the square root is: `sqrt(x^(4x-6y) + y^(4x-6y))`. And that's the expression simplified while using the "undefined" language code as required.

Understanding and Handling a Polynomial Expression

The image shows a mathematical expression which reads: \( 64y^{10} + 17y^5 + 11 \) To "solve" this expression could mean different things depending on the context, since this is just a polynomial expression. If you're looking to simplify it, it's already in simplified form, as there are no like terms to combine. If you're trying to factor it or set it equal to something and solve for \( y \), more information or a specific instruction is needed. With the information given, there is not much else to do with the expression as it stands. If you were to set this expression equal to 0, you would be looking for the roots of the polynomial equation \( 64y^{10} + 17y^5 + 11 = 0 \), but that would generally require numerical methods or more advanced algebra to solve, as it doesn't have obvious factors that would allow for easy factoring. Please provide more specific instructions if you're looking to do something else with this expression.

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