Example Question - exponent rules
Here are examples of questions we've helped users solve.
Determining the Value of x in an Exponential Expression
\[ \begin{align*} \frac{a^3 \cdot b^x \cdot c^4}{a^{1/2} \cdot b^{5/2} \cdot c^5} &= a^2 \cdot c^{-1} \\ a^{3 - \frac{1}{2}} \cdot b^{x - \frac{5}{2}} \cdot c^{4 - 5} &= a^2 \cdot c^{-1} \\ a^{\frac{5}{2}} \cdot b^{x - \frac{5}{2}} \cdot c^{-1} &= a^2 \cdot c^{-1} \\ \end{align*} \] Kita dapat memisahkan persamaan berdasarkan basis variabel a, b, dan c untuk membuatnya lebih mudah diselesaikan. \[ \begin{align*} a^{\frac{5}{2}} &= a^2 \\ b^{x - \frac{5}{2}} &= 1 \\ c^{-1} &= c^{-1} \\ \end{align*} \] Dari persamaan tersebut dapat disimpulkan bahwa: \[ \begin{align*} \frac{5}{2} &= 2 \quad \text{(Ini benar, tidak perlu dipecahkan lebih lanjut)} \\ x - \frac{5}{2} &= 0 \quad \text{(Ini yang akan kita selesaikan)} \\ \end{align*} \] \[ \begin{align*} x - \frac{5}{2} &= 0 \\ x &= \frac{5}{2} \\ \end{align*} \]
Exponent Rules Applied to Powers of Two
<p>Langkah pertama, kita akan menggunakan aturan pangkat saat mengalikan bilangan dengan basis yang sama:</p> <p>Jika \(a^m \times a^n = a^{m+n}\), maka:</p> <p>\(2^2 \times 2^3 = 2^{2+3}\)</p> <p>Langkah kedua, kita menjumlahkan eksponennya:</p> <p>\(2^{2+3} = 2^5\)</p> <p>Langkah ketiga, kita menghitung nilai \(2^5\):</p> <p>\(2^5 = 32\)</p>
Simplifying a Power Expression
<p>Para simplificar la expresión \( x^7 x^{-n} \), se aplican las reglas de los exponentes, las cuales indican que al multiplicar dos expresiones con la misma base, los exponentes se suman.</p> <p>\( x^7 \cdot x^{-n} = x^{7 + (-n)} = x^{7-n} \)</p> <p>Por lo tanto, la expresión simplificada es \( x^{7-n} \).</p>
Solving an Exponential Expression
<p>Данное выражение: \( 4^{log_3{2}} \)</p> <p>По определению логарифма \( a^{log_a{b}} = b \), поэтому:</p> <p>\( 4^{log_3{2}} = 2^{log_3{4}} \)</p> <p>Используем свойство логарифмов \( log_a{b^n} = n \cdot log_a{b} \) и выразим \( log_3{4} \) как \( 2 \cdot log_3{2} \):</p> <p>\( 2^{log_3{4}} = 2^{2 \cdot log_3{2}} \)</p> <p>Тогда используем свойство экспонент: \( a^{m \cdot n} = (a^m)^n \):</p> <p>\( 2^{2 \cdot log_3{2}} = (2^{log_3{2}})^2 \)</p> <p>По определению логарифма возводим обратно:</p> <p>\( (3^{log_3{2}})^2 = 2^2 \)</p> <p>Таким образом, итоговый ответ:</p> <p>\( 2^2 = 4 \)</p>
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 a Fraction with Exponents
Given the expression \(\frac{4a^6b^5c^{-2}}{(2a^{-4}b^6c)^3}\), simplify as follows: \[ \left(\frac{4a^6b^5c^{-2}}{(2a^{-4}b^6c)^3}\right) = \frac{4a^6b^5c^{-2}}{8a^{-12}b^{18}c^3} \] Apply the exponent rule \(a^m \cdot a^n = a^{m+n}\) and simplify: \[ = 4a^{6 - (-12)}b^{5 - 18}c^{-2 - 3} \] \[ = 4a^{18}b^{-13}c^{-5} \] \[ = \frac{4a^{18}}{b^{13}c^{5}} \] Thus, the final simplified form is \(\frac{4a^{18}}{b^{13}c^{5}}\).
Simplifying Expression with Radical Exponents
The image is rotated and not fully clear, but I will provide the steps for the equation that is visible: The problem looks like it involves simplifying the expression 2·3^(1/5) / (6^(2/5)·(3^(1/5))²) Step 1: Compute the exponent for 3^(1/5) when squared, which is (1/5) * 2 = 2/5. Step 2: Simplify the expression as such: 2·3^(1/5) / (6^(2/5)·3^(2/5)) Since 6 = 2·3, we can rewrite 6^(2/5) as (2·3)^(2/5), which is equivalent to 2^(2/5)·3^(2/5). 3: The expression is now: 2·3^(1/5) / (2^(2/5)·3^(2/5)) 4: We can now cancel out common terms: (2·3^(1/5)) / (2^(2/5)·3^(2/5)) = 3^(1/5) / 2^(2/5) 5: The solution will be the simplified form of 3^(1/5) / 2^(2/5). If you can provide a clearer image or clarify the exact expression, I can offer more precise help.
Solving a Negative Exponent Expression
Para resolver esta expresión con exponente negativo, debes recordar que una base elevada a un exponente negativo es equivalente a la inversa de la base elevada al exponente positivo. El 3/5 es la base y -2 es el exponente. Entonces, puedes reescribir la expresión tomando el recíproco de 3/5 y cambiando el signo del exponente a positivo: \[ \left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^{2} \] Ahora elevamos al cuadrado el nuevo fraccionario 5/3, lo cual significa que multiplicamos 5/3 por sí mismo: \[ \left(\frac{5}{3}\right)^{2} = \frac{5}{3} \cdot \frac{5}{3} = \frac{5 \cdot 5}{3 \cdot 3} = \frac{25}{9} \] Por lo tanto, la solución final de la expresión es \( \frac{25}{9} \).
Simplified Expression Calculation
The expression given in the image is: \((x^2y^5 ÷ y^0)^2\) Firstly, let's simplify the expression inside the parentheses. Any nonzero number raised to the power of 0 is 1, which means \(y^0 = 1\). Therefore, our expression becomes: \((x^2y^5 ÷ 1)^2\) Since dividing by 1 does not change the value of the expression, we have: \((x^2y^5)^2\) Next, when you raise a power to another power, you multiply the exponents. Here's how to break it down: \((x^2)^2 * (y^5)^2\) Now calculate each term: \(x^2\) raised to the 2nd power is \(x^{2*2}\) which is \(x^4\), and \(y^5\) raised to the 2nd power is \(y^{5*2}\) which is \(y^{10}\). So after combining them, you get: \(x^4y^{10}\) This is the simplified form of the original expression.
Simplifying Exponential Expression
The expression given in the image is: \[ \left( \frac{8y^4z^8}{16y^8z} \right)^4 \] To simplify this expression, first, simplify the fraction by canceling common factors and then apply the exponent of 4: \[ \left( \frac{8}{16} \cdot \frac{y^4}{y^8} \cdot \frac{z^8}{z} \right)^4 \] Simplify the fractions: \[ \left( \frac{1}{2} \cdot y^{4-8} \cdot z^{8-1} \right)^4 \] which simplifies further to: \[ \left( \frac{1}{2} \cdot y^{-4} \cdot z^7 \right)^4 \] Now apply the exponent of 4 to each term within the parentheses: \[ \left( \frac{1}{2} \right)^4 \cdot y^{-4 \cdot 4} \cdot z^{7 \cdot 4} \] This gives: \[ \frac{1}{16} \cdot y^{-16} \cdot z^{28} \] Since y has a negative exponent, it can be moved to the denominator: \[ \frac{z^{28}}{16y^{16}} \] This is the simplified form of the original expression.
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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